TPTP Problem File: ITP254_3.p
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%------------------------------------------------------------------------------
% File : ITP254_3 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Bounds 01696_095800
% 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_01696_095800 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 13018 (4293 unt;2245 typ; 0 def)
% Number of atoms : 26643 (9785 equ)
% Maximal formula atoms : 73 ( 2 avg)
% Number of connectives : 17416 (2542 ~; 511 |;1314 &)
% (2731 <=>;10318 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 5 avg)
% Maximal term depth : 23 ( 2 avg)
% Number of FOOLs : 1510 ( 733 fml; 777 var)
% Number of X terms : 1028 ( 0 []; 831 ite; 197 let)
% Number of types : 429 ( 428 usr)
% Number of type conns : 2045 (1371 >; 674 *; 0 +; 0 <<)
% Number of predicates : 107 ( 104 usr; 2 prp; 0-4 aty)
% Number of functors : 1731 (1731 usr; 464 con; 0-8 aty)
% Number of variables : 26714 (25846 !; 868 ?;26714 :)
% SPC : TX0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-18 04:16:46.596
%------------------------------------------------------------------------------
% Could-be-implicit typings (428)
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tff(sy_c_ATP_058Lamm__ap____,type,
aTP_Lamm_ap: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__aq____,type,
aTP_Lamm_aq: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ar____,type,
aTP_Lamm_ar: fun_int_int ).
tff(sy_c_ATP_058Lamm__as____,type,
aTP_Lamm_as: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__at____,type,
aTP_Lamm_at: rat > fun_rat_rat ).
tff(sy_c_ATP_058Lamm__au____,type,
aTP_Lamm_au: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__av____,type,
aTP_Lamm_av: int > fun_int_int ).
tff(sy_c_ATP_058Lamm__aw____,type,
aTP_Lamm_aw: fun_nat_o > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__ax____,type,
aTP_Lamm_ax: fun_nat_nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__ay____,type,
aTP_Lamm_ay: set_complex > fun_na2418085665696872764plex_o ).
tff(sy_c_ATP_058Lamm__az____,type,
aTP_Lamm_az: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).
tff(sy_c_ATP_058Lamm__ba____,type,
aTP_Lamm_ba: set_o > fun_nat_fun_list_o_o ).
tff(sy_c_ATP_058Lamm__bb____,type,
aTP_Lamm_bb: set_int > fun_na7306406234920378942_int_o ).
tff(sy_c_ATP_058Lamm__bc____,type,
aTP_Lamm_bc: set_nat > fun_na7234455742697417626_nat_o ).
tff(sy_c_ATP_058Lamm__bd____,type,
aTP_Lamm_bd: set_complex > fun_na2418085665696872764plex_o ).
tff(sy_c_ATP_058Lamm__be____,type,
aTP_Lamm_be: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).
tff(sy_c_ATP_058Lamm__bf____,type,
aTP_Lamm_bf: set_o > fun_nat_fun_list_o_o ).
tff(sy_c_ATP_058Lamm__bg____,type,
aTP_Lamm_bg: set_int > fun_na7306406234920378942_int_o ).
tff(sy_c_ATP_058Lamm__bh____,type,
aTP_Lamm_bh: set_nat > fun_na7234455742697417626_nat_o ).
tff(sy_c_ATP_058Lamm__bi____,type,
aTP_Lamm_bi: fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__bj____,type,
aTP_Lamm_bj: fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__bk____,type,
aTP_Lamm_bk: int > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__bl____,type,
aTP_Lamm_bl: int > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__bm____,type,
aTP_Lamm_bm: int > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__bn____,type,
aTP_Lamm_bn: nat > fun_co1148497455491383662plex_o ).
tff(sy_c_ATP_058Lamm__bo____,type,
aTP_Lamm_bo: nat > fun_real_o ).
tff(sy_c_ATP_058Lamm__bp____,type,
aTP_Lamm_bp: nat > fun_complex_o ).
tff(sy_c_ATP_058Lamm__bq____,type,
aTP_Lamm_bq: set_real > fun_fu7444430165986413063real_o ).
tff(sy_c_ATP_058Lamm__br____,type,
aTP_Lamm_br: set_real > fun_fu9068519094125542729real_o ).
tff(sy_c_ATP_058Lamm__bs____,type,
aTP_Lamm_bs: set_nat > fun_fu6176141739774794687_nat_o ).
tff(sy_c_ATP_058Lamm__bt____,type,
aTP_Lamm_bt: set_nat > fun_fu6433034133342731997_nat_o ).
tff(sy_c_ATP_058Lamm__bu____,type,
aTP_Lamm_bu: set_int > fun_fu7994259084669964039_int_o ).
tff(sy_c_ATP_058Lamm__bv____,type,
aTP_Lamm_bv: set_int > fun_fu3930813892005118153_int_o ).
tff(sy_c_ATP_058Lamm__bw____,type,
aTP_Lamm_bw: set_complex > fun_fu2445682883699234307plex_o ).
tff(sy_c_ATP_058Lamm__bx____,type,
aTP_Lamm_bx: set_complex > fun_fu2142003765521078979plex_o ).
tff(sy_c_ATP_058Lamm__by____,type,
aTP_Lamm_by: set_real > fun_fu3022329567491847689real_o ).
tff(sy_c_ATP_058Lamm__bz____,type,
aTP_Lamm_bz: set_real > fun_fu5533088652784741581real_o ).
tff(sy_c_ATP_058Lamm__ca____,type,
aTP_Lamm_ca: set_nat > fun_fu6621322437453427393_nat_o ).
tff(sy_c_ATP_058Lamm__cb____,type,
aTP_Lamm_cb: set_nat > fun_fu6813129056075673441_nat_o ).
tff(sy_c_ATP_058Lamm__cc____,type,
aTP_Lamm_cc: set_int > fun_fu4185471554550717449_int_o ).
tff(sy_c_ATP_058Lamm__cd____,type,
aTP_Lamm_cd: set_int > fun_fu4228744305898656589_int_o ).
tff(sy_c_ATP_058Lamm__ce____,type,
aTP_Lamm_ce: set_complex > fun_fu806425790188207109plex_o ).
tff(sy_c_ATP_058Lamm__cf____,type,
aTP_Lamm_cf: set_complex > fun_fu7523448966510524487plex_o ).
tff(sy_c_ATP_058Lamm__cg____,type,
aTP_Lamm_cg: set_real > fun_fu6510078244439976541real_o ).
tff(sy_c_ATP_058Lamm__ch____,type,
aTP_Lamm_ch: set_real > fun_fu5587483295140630005real_o ).
tff(sy_c_ATP_058Lamm__ci____,type,
aTP_Lamm_ci: set_nat > fun_fu7145897819039579925_nat_o ).
tff(sy_c_ATP_058Lamm__cj____,type,
aTP_Lamm_cj: set_nat > fun_fu1806770557662746761_nat_o ).
tff(sy_c_ATP_058Lamm__ck____,type,
aTP_Lamm_ck: set_real > fun_fu7444430165986413063real_o ).
tff(sy_c_ATP_058Lamm__cl____,type,
aTP_Lamm_cl: set_real > fun_fu9068519094125542729real_o ).
tff(sy_c_ATP_058Lamm__cm____,type,
aTP_Lamm_cm: set_nat > fun_fu6176141739774794687_nat_o ).
tff(sy_c_ATP_058Lamm__cn____,type,
aTP_Lamm_cn: set_nat > fun_fu6433034133342731997_nat_o ).
tff(sy_c_ATP_058Lamm__co____,type,
aTP_Lamm_co: set_int > fun_fu7994259084669964039_int_o ).
tff(sy_c_ATP_058Lamm__cp____,type,
aTP_Lamm_cp: set_int > fun_fu3930813892005118153_int_o ).
tff(sy_c_ATP_058Lamm__cq____,type,
aTP_Lamm_cq: set_complex > fun_fu2445682883699234307plex_o ).
tff(sy_c_ATP_058Lamm__cr____,type,
aTP_Lamm_cr: set_complex > fun_fu2142003765521078979plex_o ).
tff(sy_c_ATP_058Lamm__cs____,type,
aTP_Lamm_cs: set_real > fun_fu3022329567491847689real_o ).
tff(sy_c_ATP_058Lamm__ct____,type,
aTP_Lamm_ct: set_real > fun_fu5533088652784741581real_o ).
tff(sy_c_ATP_058Lamm__cu____,type,
aTP_Lamm_cu: set_nat > fun_fu6621322437453427393_nat_o ).
tff(sy_c_ATP_058Lamm__cv____,type,
aTP_Lamm_cv: set_nat > fun_fu6813129056075673441_nat_o ).
tff(sy_c_ATP_058Lamm__cw____,type,
aTP_Lamm_cw: set_int > fun_fu4185471554550717449_int_o ).
tff(sy_c_ATP_058Lamm__cx____,type,
aTP_Lamm_cx: set_int > fun_fu4228744305898656589_int_o ).
tff(sy_c_ATP_058Lamm__cy____,type,
aTP_Lamm_cy: set_complex > fun_fu806425790188207109plex_o ).
tff(sy_c_ATP_058Lamm__cz____,type,
aTP_Lamm_cz: set_complex > fun_fu7523448966510524487plex_o ).
tff(sy_c_ATP_058Lamm__da____,type,
aTP_Lamm_da: set_real > fun_fu6510078244439976541real_o ).
tff(sy_c_ATP_058Lamm__db____,type,
aTP_Lamm_db: set_real > fun_fu5587483295140630005real_o ).
tff(sy_c_ATP_058Lamm__dc____,type,
aTP_Lamm_dc: set_nat > fun_fu7145897819039579925_nat_o ).
tff(sy_c_ATP_058Lamm__dd____,type,
aTP_Lamm_dd: set_nat > fun_fu1806770557662746761_nat_o ).
tff(sy_c_ATP_058Lamm__de____,type,
aTP_Lamm_de: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__df____,type,
aTP_Lamm_df: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__dg____,type,
aTP_Lamm_dg: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__dh____,type,
aTP_Lamm_dh: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__di____,type,
aTP_Lamm_di: fun_nat_nat > fun_fu1217771548038759377at_nat ).
tff(sy_c_ATP_058Lamm__dj____,type,
aTP_Lamm_dj: fun_o_o > fun_fu138107234227827011_int_o ).
tff(sy_c_ATP_058Lamm__dk____,type,
aTP_Lamm_dk: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).
tff(sy_c_ATP_058Lamm__dl____,type,
aTP_Lamm_dl: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).
tff(sy_c_ATP_058Lamm__dm____,type,
aTP_Lamm_dm: fun_na3970208099681795815_nat_o > fun_fu484727072080742171_nat_o ).
tff(sy_c_ATP_058Lamm__dn____,type,
aTP_Lamm_dn: fun_fu1115759055147092071_o_nat > fun_fu4228150600790569371at_nat ).
tff(sy_c_ATP_058Lamm__do____,type,
aTP_Lamm_do: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).
tff(sy_c_ATP_058Lamm__dp____,type,
aTP_Lamm_dp: fun_na8713714802964096592at_nat > fun_fu1039311746440044420at_nat ).
tff(sy_c_ATP_058Lamm__dq____,type,
aTP_Lamm_dq: fun_fu1413608398870722896at_nat > fun_fu7448223259553364356at_nat ).
tff(sy_c_ATP_058Lamm__dr____,type,
aTP_Lamm_dr: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).
tff(sy_c_ATP_058Lamm__ds____,type,
aTP_Lamm_ds: fun_Pr2602117396471556196at_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__dt____,type,
aTP_Lamm_dt: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__du____,type,
aTP_Lamm_du: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).
tff(sy_c_ATP_058Lamm__dv____,type,
aTP_Lamm_dv: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__dw____,type,
aTP_Lamm_dw: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__dx____,type,
aTP_Lamm_dx: fun_co1148497455491383662plex_o ).
tff(sy_c_ATP_058Lamm__dy____,type,
aTP_Lamm_dy: fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__dz____,type,
aTP_Lamm_dz: fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ea____,type,
aTP_Lamm_ea: fun_Co2081214437782444048eger_o ).
tff(sy_c_ATP_058Lamm__eb____,type,
aTP_Lamm_eb: num > fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__ec____,type,
aTP_Lamm_ec: num > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__ed____,type,
aTP_Lamm_ed: num > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__ee____,type,
aTP_Lamm_ee: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__ef____,type,
aTP_Lamm_ef: fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__eg____,type,
aTP_Lamm_eg: $o > fun_fu138107234227827011_int_o ).
tff(sy_c_ATP_058Lamm__eh____,type,
aTP_Lamm_eh: fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ei____,type,
aTP_Lamm_ei: int > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ej____,type,
aTP_Lamm_ej: int > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ek____,type,
aTP_Lamm_ek: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).
tff(sy_c_ATP_058Lamm__el____,type,
aTP_Lamm_el: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).
tff(sy_c_ATP_058Lamm__em____,type,
aTP_Lamm_em: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).
tff(sy_c_ATP_058Lamm__en____,type,
aTP_Lamm_en: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__eo____,type,
aTP_Lamm_eo: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ep____,type,
aTP_Lamm_ep: fun_int_int > fun_int_int ).
tff(sy_c_ATP_058Lamm__eq____,type,
aTP_Lamm_eq: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__er____,type,
aTP_Lamm_er: int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__es____,type,
aTP_Lamm_es: complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__et____,type,
aTP_Lamm_et: nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__eu____,type,
aTP_Lamm_eu: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__ev____,type,
aTP_Lamm_ev: fun_int_int > fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__ew____,type,
aTP_Lamm_ew: fun_complex_complex > fun_co7673397065115134084omplex ).
tff(sy_c_ATP_058Lamm__ex____,type,
aTP_Lamm_ex: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ey____,type,
aTP_Lamm_ey: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ez____,type,
aTP_Lamm_ez: fun_int_int > fun_fu923337204662071120nt_int ).
tff(sy_c_ATP_058Lamm__fa____,type,
aTP_Lamm_fa: fun_int_int > fun_fu3739269683040691718nt_int ).
tff(sy_c_ATP_058Lamm__fb____,type,
aTP_Lamm_fb: fun_complex_complex > fun_fu8998703528969179354omplex ).
tff(sy_c_ATP_058Lamm__fc____,type,
aTP_Lamm_fc: fun_complex_complex > fun_fu8084223252616112272omplex ).
tff(sy_c_ATP_058Lamm__fd____,type,
aTP_Lamm_fd: fun_nat_nat > fun_fu4413992040155564420at_nat ).
tff(sy_c_ATP_058Lamm__fe____,type,
aTP_Lamm_fe: fun_nat_nat > fun_fu2454420574033104442at_nat ).
tff(sy_c_ATP_058Lamm__ff____,type,
aTP_Lamm_ff: fun_nat_real > fun_fu163975356400620604t_real ).
tff(sy_c_ATP_058Lamm__fg____,type,
aTP_Lamm_fg: fun_nat_real > fun_fu3306995184869731826t_real ).
tff(sy_c_ATP_058Lamm__fh____,type,
aTP_Lamm_fh: fun_int_int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__fi____,type,
aTP_Lamm_fi: fun_complex_complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__fj____,type,
aTP_Lamm_fj: fun_nat_nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__fk____,type,
aTP_Lamm_fk: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__fl____,type,
aTP_Lamm_fl: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).
tff(sy_c_ATP_058Lamm__fm____,type,
aTP_Lamm_fm: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).
tff(sy_c_ATP_058Lamm__fn____,type,
aTP_Lamm_fn: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).
tff(sy_c_ATP_058Lamm__fo____,type,
aTP_Lamm_fo: fun_int_Code_integer > fun_fu3628599891403802633nteger ).
tff(sy_c_ATP_058Lamm__fp____,type,
aTP_Lamm_fp: fun_complex_rat > fun_fu2990765671291504771ex_rat ).
tff(sy_c_ATP_058Lamm__fq____,type,
aTP_Lamm_fq: fun_real_rat > fun_fu2598743018846156927al_rat ).
tff(sy_c_ATP_058Lamm__fr____,type,
aTP_Lamm_fr: fun_nat_rat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__fs____,type,
aTP_Lamm_fs: fun_int_rat > fun_fu1008360527747082367nt_rat ).
tff(sy_c_ATP_058Lamm__ft____,type,
aTP_Lamm_ft: fun_complex_int > fun_fu4213752561377164075ex_int ).
tff(sy_c_ATP_058Lamm__fu____,type,
aTP_Lamm_fu: fun_real_int > fun_fu3737244727311182119al_int ).
tff(sy_c_ATP_058Lamm__fv____,type,
aTP_Lamm_fv: fun_nat_complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__fw____,type,
aTP_Lamm_fw: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__fx____,type,
aTP_Lamm_fx: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__fy____,type,
aTP_Lamm_fy: fun_nat_complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__fz____,type,
aTP_Lamm_fz: fun_nat_rat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__ga____,type,
aTP_Lamm_ga: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__gb____,type,
aTP_Lamm_gb: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).
tff(sy_c_ATP_058Lamm__gc____,type,
aTP_Lamm_gc: fun_complex_nat > fun_fu705315015666092147ex_nat ).
tff(sy_c_ATP_058Lamm__gd____,type,
aTP_Lamm_gd: fun_real_nat > fun_fu3214020956574444655al_nat ).
tff(sy_c_ATP_058Lamm__ge____,type,
aTP_Lamm_ge: fun_int_nat > fun_fu8905809029841901679nt_nat ).
tff(sy_c_ATP_058Lamm__gf____,type,
aTP_Lamm_gf: fun_nat_nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__gg____,type,
aTP_Lamm_gg: fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__gh____,type,
aTP_Lamm_gh: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__gi____,type,
aTP_Lamm_gi: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__gj____,type,
aTP_Lamm_gj: fun_nat_real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__gk____,type,
aTP_Lamm_gk: complex > fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__gl____,type,
aTP_Lamm_gl: rat > fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__gm____,type,
aTP_Lamm_gm: int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__gn____,type,
aTP_Lamm_gn: real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__go____,type,
aTP_Lamm_go: fun_nat_nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__gp____,type,
aTP_Lamm_gp: fun_nat_real > fun_na7718645245721260691t_real ).
tff(sy_c_ATP_058Lamm__gq____,type,
aTP_Lamm_gq: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__gr____,type,
aTP_Lamm_gr: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__gs____,type,
aTP_Lamm_gs: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__gt____,type,
aTP_Lamm_gt: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__gu____,type,
aTP_Lamm_gu: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__gv____,type,
aTP_Lamm_gv: fun_nat_complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__gw____,type,
aTP_Lamm_gw: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__gx____,type,
aTP_Lamm_gx: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__gy____,type,
aTP_Lamm_gy: fun_nat_real > fun_nat_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_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__hc____,type,
aTP_Lamm_hc: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hd____,type,
aTP_Lamm_hd: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__he____,type,
aTP_Lamm_he: fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__hf____,type,
aTP_Lamm_hf: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__hg____,type,
aTP_Lamm_hg: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__hh____,type,
aTP_Lamm_hh: vEBT_VEBT > fun_fu4141963935195952137T_real ).
tff(sy_c_ATP_058Lamm__hi____,type,
aTP_Lamm_hi: complex > fun_fu709861064203714431x_real ).
tff(sy_c_ATP_058Lamm__hj____,type,
aTP_Lamm_hj: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).
tff(sy_c_ATP_058Lamm__hk____,type,
aTP_Lamm_hk: complex > fun_fu6638204120336560003ex_rat ).
tff(sy_c_ATP_058Lamm__hl____,type,
aTP_Lamm_hl: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).
tff(sy_c_ATP_058Lamm__hm____,type,
aTP_Lamm_hm: complex > fun_fu2513798379308225899ex_nat ).
tff(sy_c_ATP_058Lamm__hn____,type,
aTP_Lamm_hn: vEBT_VEBT > fun_fu7059288016784409353BT_int ).
tff(sy_c_ATP_058Lamm__ho____,type,
aTP_Lamm_ho: complex > fun_fu4692242605864086655ex_int ).
tff(sy_c_ATP_058Lamm__hp____,type,
aTP_Lamm_hp: int > fun_fu4328765270694171897t_real ).
tff(sy_c_ATP_058Lamm__hq____,type,
aTP_Lamm_hq: int > fun_fu1617438229336688381nt_rat ).
tff(sy_c_ATP_058Lamm__hr____,type,
aTP_Lamm_hr: rat > fun_rat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hs____,type,
aTP_Lamm_hs: extended_enat > fun_Ex2441982661509597004d_enat ).
tff(sy_c_ATP_058Lamm__ht____,type,
aTP_Lamm_ht: complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__hu____,type,
aTP_Lamm_hu: int > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__hv____,type,
aTP_Lamm_hv: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__hw____,type,
aTP_Lamm_hw: real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__hx____,type,
aTP_Lamm_hx: fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hy____,type,
aTP_Lamm_hy: fun_na7542918316964655096d_enat ).
tff(sy_c_ATP_058Lamm__hz____,type,
aTP_Lamm_hz: fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__ia____,type,
aTP_Lamm_ia: fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__ib____,type,
aTP_Lamm_ib: fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__ic____,type,
aTP_Lamm_ic: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__id____,type,
aTP_Lamm_id: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ie____,type,
aTP_Lamm_ie: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__if____,type,
aTP_Lamm_if: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ig____,type,
aTP_Lamm_ig: fun_nat_real ).
tff(sy_c_ATP_058Lamm__ih____,type,
aTP_Lamm_ih: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__ii____,type,
aTP_Lamm_ii: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__ij____,type,
aTP_Lamm_ij: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__ik____,type,
aTP_Lamm_ik: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__il____,type,
aTP_Lamm_il: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__im____,type,
aTP_Lamm_im: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__in____,type,
aTP_Lamm_in: complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__io____,type,
aTP_Lamm_io: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ip____,type,
aTP_Lamm_ip: fun_nat_int > fun_fu8464300013002972271at_int ).
tff(sy_c_ATP_058Lamm__iq____,type,
aTP_Lamm_iq: fun_rat_fun_rat_o ).
tff(sy_c_ATP_058Lamm__ir____,type,
aTP_Lamm_ir: fun_num_fun_num_o ).
tff(sy_c_ATP_058Lamm__is____,type,
aTP_Lamm_is: fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__it____,type,
aTP_Lamm_it: fun_real_fun_real_o ).
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: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__iw____,type,
aTP_Lamm_iw: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ix____,type,
aTP_Lamm_ix: fun_nat_complex > fun_co8378349014452122532t_real ).
tff(sy_c_ATP_058Lamm__iy____,type,
aTP_Lamm_iy: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__iz____,type,
aTP_Lamm_iz: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ja____,type,
aTP_Lamm_ja: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__jb____,type,
aTP_Lamm_jb: fun_nat_complex > fun_na6632006986671646835omplex ).
tff(sy_c_ATP_058Lamm__jc____,type,
aTP_Lamm_jc: fun_nat_real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__jd____,type,
aTP_Lamm_jd: fun_nat_complex > fun_nat_real ).
tff(sy_c_ATP_058Lamm__je____,type,
aTP_Lamm_je: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__jf____,type,
aTP_Lamm_jf: fun_nat_real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__jg____,type,
aTP_Lamm_jg: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__jh____,type,
aTP_Lamm_jh: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__ji____,type,
aTP_Lamm_ji: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__jj____,type,
aTP_Lamm_jj: fun_nat_rat > fun_rat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__jk____,type,
aTP_Lamm_jk: fun_nat_int > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__jl____,type,
aTP_Lamm_jl: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__jm____,type,
aTP_Lamm_jm: fun_nat_complex > fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__jn____,type,
aTP_Lamm_jn: real > fun_fu7631928937237476593t_real ).
tff(sy_c_ATP_058Lamm__jo____,type,
aTP_Lamm_jo: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__jp____,type,
aTP_Lamm_jp: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__jq____,type,
aTP_Lamm_jq: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__jr____,type,
aTP_Lamm_jr: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__js____,type,
aTP_Lamm_js: int > fun_in6793278964266839919at_int ).
tff(sy_c_ATP_058Lamm__jt____,type,
aTP_Lamm_jt: int > fun_in6793278964266839919at_int ).
tff(sy_c_ATP_058Lamm__ju____,type,
aTP_Lamm_ju: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__jv____,type,
aTP_Lamm_jv: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__jw____,type,
aTP_Lamm_jw: complex > fun_na6632006986671646835omplex ).
tff(sy_c_ATP_058Lamm__jx____,type,
aTP_Lamm_jx: rat > fun_na1148497095348402375at_rat ).
tff(sy_c_ATP_058Lamm__jy____,type,
aTP_Lamm_jy: int > fun_na2382631619178260847at_int ).
tff(sy_c_ATP_058Lamm__jz____,type,
aTP_Lamm_jz: real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__ka____,type,
aTP_Lamm_ka: complex > fun_na6632006986671646835omplex ).
tff(sy_c_ATP_058Lamm__kb____,type,
aTP_Lamm_kb: rat > fun_na1148497095348402375at_rat ).
tff(sy_c_ATP_058Lamm__kc____,type,
aTP_Lamm_kc: int > fun_na2382631619178260847at_int ).
tff(sy_c_ATP_058Lamm__kd____,type,
aTP_Lamm_kd: real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__ke____,type,
aTP_Lamm_ke: complex > fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__kf____,type,
aTP_Lamm_kf: rat > fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__kg____,type,
aTP_Lamm_kg: int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__kh____,type,
aTP_Lamm_kh: real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__ki____,type,
aTP_Lamm_ki: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__kj____,type,
aTP_Lamm_kj: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__kk____,type,
aTP_Lamm_kk: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__kl____,type,
aTP_Lamm_kl: fun_nat_real ).
tff(sy_c_ATP_058Lamm__km____,type,
aTP_Lamm_km: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kn____,type,
aTP_Lamm_kn: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ko____,type,
aTP_Lamm_ko: real > fun_fu3477889739401271834t_real ).
tff(sy_c_ATP_058Lamm__kp____,type,
aTP_Lamm_kp: real > fun_fu5589980185390252312t_real ).
tff(sy_c_ATP_058Lamm__kq____,type,
aTP_Lamm_kq: real > fun_fu4806068833880117316t_real ).
tff(sy_c_ATP_058Lamm__kr____,type,
aTP_Lamm_kr: real > fun_fu2115079063411676732t_real ).
tff(sy_c_ATP_058Lamm__ks____,type,
aTP_Lamm_ks: real > fun_fu4540635375566031512t_real ).
tff(sy_c_ATP_058Lamm__kt____,type,
aTP_Lamm_kt: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__ku____,type,
aTP_Lamm_ku: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kv____,type,
aTP_Lamm_kv: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kw____,type,
aTP_Lamm_kw: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kx____,type,
aTP_Lamm_kx: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__ky____,type,
aTP_Lamm_ky: complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__kz____,type,
aTP_Lamm_kz: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__la____,type,
aTP_Lamm_la: fun_nat_complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__lb____,type,
aTP_Lamm_lb: nat > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__lc____,type,
aTP_Lamm_lc: nat > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__ld____,type,
aTP_Lamm_ld: nat > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__le____,type,
aTP_Lamm_le: nat > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lf____,type,
aTP_Lamm_lf: nat > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__lg____,type,
aTP_Lamm_lg: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lh____,type,
aTP_Lamm_lh: fun_nat_real > fun_se2100927751953696370t_real ).
tff(sy_c_ATP_058Lamm__li____,type,
aTP_Lamm_li: fun_nat_real ).
tff(sy_c_ATP_058Lamm__lj____,type,
aTP_Lamm_lj: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__lk____,type,
aTP_Lamm_lk: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__ll____,type,
aTP_Lamm_ll: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lm____,type,
aTP_Lamm_lm: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ln____,type,
aTP_Lamm_ln: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__lo____,type,
aTP_Lamm_lo: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lp____,type,
aTP_Lamm_lp: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lq____,type,
aTP_Lamm_lq: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lr____,type,
aTP_Lamm_lr: complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__ls____,type,
aTP_Lamm_ls: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__lt____,type,
aTP_Lamm_lt: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__lu____,type,
aTP_Lamm_lu: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lv____,type,
aTP_Lamm_lv: complex > fun_na1591161823130732966omplex ).
tff(sy_c_ATP_058Lamm__lw____,type,
aTP_Lamm_lw: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__lx____,type,
aTP_Lamm_lx: int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__ly____,type,
aTP_Lamm_ly: real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__lz____,type,
aTP_Lamm_lz: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ma____,type,
aTP_Lamm_ma: fun_complex_complex ).
tff(sy_c_ATP_058Lamm__mb____,type,
aTP_Lamm_mb: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__mc____,type,
aTP_Lamm_mc: fun_int_int ).
tff(sy_c_ATP_058Lamm__md____,type,
aTP_Lamm_md: fun_real_real ).
tff(sy_c_ATP_058Lamm__me____,type,
aTP_Lamm_me: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__mf____,type,
aTP_Lamm_mf: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__mg____,type,
aTP_Lamm_mg: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__mh____,type,
aTP_Lamm_mh: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__mi____,type,
aTP_Lamm_mi: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__mj____,type,
aTP_Lamm_mj: fun_nat_int ).
tff(sy_c_ATP_058Lamm__mk____,type,
aTP_Lamm_mk: fun_int_int ).
tff(sy_c_ATP_058Lamm__ml____,type,
aTP_Lamm_ml: real > fun_fu8384527901749275179omplex ).
tff(sy_c_ATP_058Lamm__mm____,type,
aTP_Lamm_mm: nat > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__mn____,type,
aTP_Lamm_mn: int > fun_fu3613808716698379563omplex ).
tff(sy_c_ATP_058Lamm__mo____,type,
aTP_Lamm_mo: complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__mp____,type,
aTP_Lamm_mp: real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__mq____,type,
aTP_Lamm_mq: nat > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__mr____,type,
aTP_Lamm_mr: int > fun_fu121957152452960295t_real ).
tff(sy_c_ATP_058Lamm__ms____,type,
aTP_Lamm_ms: complex > fun_fu7714627026455246123x_real ).
tff(sy_c_ATP_058Lamm__mt____,type,
aTP_Lamm_mt: real > fun_fu2598743018846156927al_rat ).
tff(sy_c_ATP_058Lamm__mu____,type,
aTP_Lamm_mu: nat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__mv____,type,
aTP_Lamm_mv: real > fun_fu8384527901749275179omplex ).
tff(sy_c_ATP_058Lamm__mw____,type,
aTP_Lamm_mw: nat > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__mx____,type,
aTP_Lamm_mx: int > fun_fu3613808716698379563omplex ).
tff(sy_c_ATP_058Lamm__my____,type,
aTP_Lamm_my: complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__mz____,type,
aTP_Lamm_mz: real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__na____,type,
aTP_Lamm_na: nat > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__nb____,type,
aTP_Lamm_nb: int > fun_fu121957152452960295t_real ).
tff(sy_c_ATP_058Lamm__nc____,type,
aTP_Lamm_nc: complex > fun_fu7714627026455246123x_real ).
tff(sy_c_ATP_058Lamm__nd____,type,
aTP_Lamm_nd: real > fun_fu2598743018846156927al_rat ).
tff(sy_c_ATP_058Lamm__ne____,type,
aTP_Lamm_ne: nat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__nf____,type,
aTP_Lamm_nf: fun_nat_nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__ng____,type,
aTP_Lamm_ng: fun_nat_int > fun_fu8464300013002972271at_int ).
tff(sy_c_ATP_058Lamm__nh____,type,
aTP_Lamm_nh: fun_int_int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__ni____,type,
aTP_Lamm_ni: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__nj____,type,
aTP_Lamm_nj: fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__nk____,type,
aTP_Lamm_nk: fun_int_int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__nl____,type,
aTP_Lamm_nl: fun_nat_complex > fun_na1591161823130732966omplex ).
tff(sy_c_ATP_058Lamm__nm____,type,
aTP_Lamm_nm: fun_nat_real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__nn____,type,
aTP_Lamm_nn: fun_nat_rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__no____,type,
aTP_Lamm_no: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__np____,type,
aTP_Lamm_np: fun_nat_int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__nq____,type,
aTP_Lamm_nq: set_real > fun_fu9050450510664235267real_o ).
tff(sy_c_ATP_058Lamm__nr____,type,
aTP_Lamm_nr: fun_real_complex > fun_fu3532930458344041007omplex ).
tff(sy_c_ATP_058Lamm__ns____,type,
aTP_Lamm_ns: set_nat > fun_fu4001317426178549763_nat_o ).
tff(sy_c_ATP_058Lamm__nt____,type,
aTP_Lamm_nt: fun_nat_complex > fun_fu4302481349826706223omplex ).
tff(sy_c_ATP_058Lamm__nu____,type,
aTP_Lamm_nu: set_int > fun_fu3883539188510166275_int_o ).
tff(sy_c_ATP_058Lamm__nv____,type,
aTP_Lamm_nv: fun_int_complex > fun_fu3558165918884697647omplex ).
tff(sy_c_ATP_058Lamm__nw____,type,
aTP_Lamm_nw: set_complex > fun_fu1358370060594899587plex_o ).
tff(sy_c_ATP_058Lamm__nx____,type,
aTP_Lamm_nx: fun_complex_complex > fun_fu3719809307426920623omplex ).
tff(sy_c_ATP_058Lamm__ny____,type,
aTP_Lamm_ny: fun_real_real > fun_fu4478067837645853357l_real ).
tff(sy_c_ATP_058Lamm__nz____,type,
aTP_Lamm_nz: fun_nat_real > fun_fu4002475321494201261t_real ).
tff(sy_c_ATP_058Lamm__oa____,type,
aTP_Lamm_oa: fun_int_real > fun_fu7030473000426307245t_real ).
tff(sy_c_ATP_058Lamm__ob____,type,
aTP_Lamm_ob: fun_complex_real > fun_fu3043978019632300333x_real ).
tff(sy_c_ATP_058Lamm__oc____,type,
aTP_Lamm_oc: fun_real_rat > fun_fu5551332503302594393al_rat ).
tff(sy_c_ATP_058Lamm__od____,type,
aTP_Lamm_od: fun_nat_rat > fun_fu3162257917499486297at_rat ).
tff(sy_c_ATP_058Lamm__oe____,type,
aTP_Lamm_oe: real > fun_fu3106432010075488275t_real ).
tff(sy_c_ATP_058Lamm__of____,type,
aTP_Lamm_of: complex > fun_fu4781654129426620053omplex ).
tff(sy_c_ATP_058Lamm__og____,type,
aTP_Lamm_og: nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__oh____,type,
aTP_Lamm_oh: int > fun_fu4512855384587262995at_int ).
tff(sy_c_ATP_058Lamm__oi____,type,
aTP_Lamm_oi: int > fun_fu6415725884910174923nt_int ).
tff(sy_c_ATP_058Lamm__oj____,type,
aTP_Lamm_oj: fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__ok____,type,
aTP_Lamm_ok: fun_int_complex > fun_int_o ).
tff(sy_c_ATP_058Lamm__ol____,type,
aTP_Lamm_ol: fun_complex_complex > fun_complex_o ).
tff(sy_c_ATP_058Lamm__om____,type,
aTP_Lamm_om: fun_int_real > fun_int_o ).
tff(sy_c_ATP_058Lamm__on____,type,
aTP_Lamm_on: fun_complex_real > fun_complex_o ).
tff(sy_c_ATP_058Lamm__oo____,type,
aTP_Lamm_oo: fun_int_rat > fun_int_o ).
tff(sy_c_ATP_058Lamm__op____,type,
aTP_Lamm_op: fun_complex_rat > fun_complex_o ).
tff(sy_c_ATP_058Lamm__oq____,type,
aTP_Lamm_oq: fun_int_nat > fun_int_o ).
tff(sy_c_ATP_058Lamm__or____,type,
aTP_Lamm_or: fun_complex_nat > fun_complex_o ).
tff(sy_c_ATP_058Lamm__os____,type,
aTP_Lamm_os: fun_complex_int > fun_complex_o ).
tff(sy_c_ATP_058Lamm__ot____,type,
aTP_Lamm_ot: fun_nat_complex > fun_nat_o ).
tff(sy_c_ATP_058Lamm__ou____,type,
aTP_Lamm_ou: fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__ov____,type,
aTP_Lamm_ov: fun_nat_int > fun_na7654559027617327379at_int ).
tff(sy_c_ATP_058Lamm__ow____,type,
aTP_Lamm_ow: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__ox____,type,
aTP_Lamm_ox: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__oy____,type,
aTP_Lamm_oy: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__oz____,type,
aTP_Lamm_oz: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__pa____,type,
aTP_Lamm_pa: int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__pb____,type,
aTP_Lamm_pb: real > fun_fu9132278742410140537l_real ).
tff(sy_c_ATP_058Lamm__pc____,type,
aTP_Lamm_pc: nat > fun_fu1620311477532340197t_real ).
tff(sy_c_ATP_058Lamm__pd____,type,
aTP_Lamm_pd: real > fun_fu7187499828406537085al_rat ).
tff(sy_c_ATP_058Lamm__pe____,type,
aTP_Lamm_pe: nat > fun_fu6579618355643354601at_rat ).
tff(sy_c_ATP_058Lamm__pf____,type,
aTP_Lamm_pf: complex > fun_na1591161823130732966omplex ).
tff(sy_c_ATP_058Lamm__pg____,type,
aTP_Lamm_pg: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__ph____,type,
aTP_Lamm_ph: real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__pi____,type,
aTP_Lamm_pi: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__pj____,type,
aTP_Lamm_pj: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__pk____,type,
aTP_Lamm_pk: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__pl____,type,
aTP_Lamm_pl: int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__pm____,type,
aTP_Lamm_pm: rat > fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__pn____,type,
aTP_Lamm_pn: real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__po____,type,
aTP_Lamm_po: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__pp____,type,
aTP_Lamm_pp: int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__pq____,type,
aTP_Lamm_pq: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__pr____,type,
aTP_Lamm_pr: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ps____,type,
aTP_Lamm_ps: fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__pt____,type,
aTP_Lamm_pt: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__pu____,type,
aTP_Lamm_pu: fun_nat_complex > fun_na1591161823130732966omplex ).
tff(sy_c_ATP_058Lamm__pv____,type,
aTP_Lamm_pv: fun_nat_rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__pw____,type,
aTP_Lamm_pw: fun_nat_int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__px____,type,
aTP_Lamm_px: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__py____,type,
aTP_Lamm_py: fun_nat_real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__pz____,type,
aTP_Lamm_pz: nat > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__qa____,type,
aTP_Lamm_qa: nat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__qb____,type,
aTP_Lamm_qb: fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__qc____,type,
aTP_Lamm_qc: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qd____,type,
aTP_Lamm_qd: set_nat > fun_set_nat_o ).
tff(sy_c_ATP_058Lamm__qe____,type,
aTP_Lamm_qe: fun_rat_fun_rat_o ).
tff(sy_c_ATP_058Lamm__qf____,type,
aTP_Lamm_qf: fun_num_fun_num_o ).
tff(sy_c_ATP_058Lamm__qg____,type,
aTP_Lamm_qg: fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__qh____,type,
aTP_Lamm_qh: fun_real_fun_real_o ).
tff(sy_c_ATP_058Lamm__qi____,type,
aTP_Lamm_qi: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qj____,type,
aTP_Lamm_qj: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qk____,type,
aTP_Lamm_qk: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ql____,type,
aTP_Lamm_ql: nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__qm____,type,
aTP_Lamm_qm: nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__qn____,type,
aTP_Lamm_qn: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qo____,type,
aTP_Lamm_qo: fun_nat_fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qp____,type,
aTP_Lamm_qp: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__qq____,type,
aTP_Lamm_qq: fun_nat_fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__qr____,type,
aTP_Lamm_qr: nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__qs____,type,
aTP_Lamm_qs: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qt____,type,
aTP_Lamm_qt: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qu____,type,
aTP_Lamm_qu: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qv____,type,
aTP_Lamm_qv: fun_nat_nat > fun_fu4413992040155564420at_nat ).
tff(sy_c_ATP_058Lamm__qw____,type,
aTP_Lamm_qw: fun_nat_nat > fun_fu4413992040155564420at_nat ).
tff(sy_c_ATP_058Lamm__qx____,type,
aTP_Lamm_qx: fun_real_fun_real_o ).
tff(sy_c_ATP_058Lamm__qy____,type,
aTP_Lamm_qy: fun_real_fun_real_o ).
tff(sy_c_ATP_058Lamm__qz____,type,
aTP_Lamm_qz: fun_real_o ).
tff(sy_c_ATP_058Lamm__ra____,type,
aTP_Lamm_ra: real > fun_int_o ).
tff(sy_c_ATP_058Lamm__rb____,type,
aTP_Lamm_rb: nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__rc____,type,
aTP_Lamm_rc: rat > fun_int_o ).
tff(sy_c_ATP_058Lamm__rd____,type,
aTP_Lamm_rd: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__re____,type,
aTP_Lamm_re: int > fun_in5570076352459579423_int_o ).
tff(sy_c_ATP_058Lamm__rf____,type,
aTP_Lamm_rf: rat > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__rg____,type,
aTP_Lamm_rg: int > fun_in5570076352459579423_int_o ).
tff(sy_c_ATP_058Lamm__rh____,type,
aTP_Lamm_rh: rat > fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ri____,type,
aTP_Lamm_ri: nat > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__rj____,type,
aTP_Lamm_rj: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__rk____,type,
aTP_Lamm_rk: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__rl____,type,
aTP_Lamm_rl: complex > fun_real_o ).
tff(sy_c_ATP_058Lamm__rm____,type,
aTP_Lamm_rm: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__rn____,type,
aTP_Lamm_rn: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__ro____,type,
aTP_Lamm_ro: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__rp____,type,
aTP_Lamm_rp: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__rq____,type,
aTP_Lamm_rq: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__rr____,type,
aTP_Lamm_rr: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__rs____,type,
aTP_Lamm_rs: complex > fun_na1554462459066888140plex_o ).
tff(sy_c_ATP_058Lamm__rt____,type,
aTP_Lamm_rt: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ru____,type,
aTP_Lamm_ru: fun_Co1092983284983208416er_int ).
tff(sy_c_ATP_058Lamm__rv____,type,
aTP_Lamm_rv: fun_Co8521658056209057742er_num ).
tff(sy_c_ATP_058Lamm__rw____,type,
aTP_Lamm_rw: fun_Co3583066429914935172er_nat ).
tff(sy_c_ATP_058Lamm__rx____,type,
aTP_Lamm_rx: fun_nat_o ).
tff(sy_c_ATP_058Lamm__ry____,type,
aTP_Lamm_ry: fun_nat_o ).
tff(sy_c_ATP_058Lamm__rz____,type,
aTP_Lamm_rz: set_nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__sa____,type,
aTP_Lamm_sa: set_nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__sb____,type,
aTP_Lamm_sb: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__sc____,type,
aTP_Lamm_sc: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__sd____,type,
aTP_Lamm_sd: code_integer > fun_Co3544761417380347684eger_o ).
tff(sy_c_ATP_058Lamm__se____,type,
aTP_Lamm_se: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__sf____,type,
aTP_Lamm_sf: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__sg____,type,
aTP_Lamm_sg: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__sh____,type,
aTP_Lamm_sh: nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__si____,type,
aTP_Lamm_si: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__sj____,type,
aTP_Lamm_sj: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__sk____,type,
aTP_Lamm_sk: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__sl____,type,
aTP_Lamm_sl: nat > fun_na7928631014926962227_nat_o ).
tff(sy_c_ATP_058Lamm__sm____,type,
aTP_Lamm_sm: fun_na5914473681949881652_nat_o ).
tff(sy_c_ATP_058Lamm__sn____,type,
aTP_Lamm_sn: nat > fun_na7928631014926962227_nat_o ).
tff(sy_c_ATP_058Lamm__so____,type,
aTP_Lamm_so: fun_na5914473681949881652_nat_o ).
tff(sy_c_ATP_058Lamm__sp____,type,
aTP_Lamm_sp: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__sq____,type,
aTP_Lamm_sq: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__sr____,type,
aTP_Lamm_sr: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__ss____,type,
aTP_Lamm_ss: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__st____,type,
aTP_Lamm_st: num > fun_num_int ).
tff(sy_c_ATP_058Lamm__su____,type,
aTP_Lamm_su: fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__sv____,type,
aTP_Lamm_sv: fun_num_option_num ).
tff(sy_c_ATP_058Lamm__sw____,type,
aTP_Lamm_sw: fun_num_option_num ).
tff(sy_c_ATP_058Lamm__sx____,type,
aTP_Lamm_sx: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__sy____,type,
aTP_Lamm_sy: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__sz____,type,
aTP_Lamm_sz: nat > fun_num_option_num ).
tff(sy_c_ATP_058Lamm__ta____,type,
aTP_Lamm_ta: nat > fun_num_option_num ).
tff(sy_c_ATP_058Lamm__tb____,type,
aTP_Lamm_tb: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__tc____,type,
aTP_Lamm_tc: fun_na7595719129879163854on_num ).
tff(sy_c_ATP_058Lamm__td____,type,
aTP_Lamm_td: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__te____,type,
aTP_Lamm_te: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__tf____,type,
aTP_Lamm_tf: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__tg____,type,
aTP_Lamm_tg: fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__th____,type,
aTP_Lamm_th: fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__ti____,type,
aTP_Lamm_ti: fun_real_real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__tj____,type,
aTP_Lamm_tj: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__tk____,type,
aTP_Lamm_tk: fun_real_real > fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__tl____,type,
aTP_Lamm_tl: fun_real_real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__tm____,type,
aTP_Lamm_tm: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__tn____,type,
aTP_Lamm_tn: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).
tff(sy_c_ATP_058Lamm__to____,type,
aTP_Lamm_to: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).
tff(sy_c_ATP_058Lamm__tp____,type,
aTP_Lamm_tp: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__tq____,type,
aTP_Lamm_tq: nat > fun_fu3612569900928539882l_real ).
tff(sy_c_ATP_058Lamm__tr____,type,
aTP_Lamm_tr: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__ts____,type,
aTP_Lamm_ts: fun_real_real ).
tff(sy_c_ATP_058Lamm__tt____,type,
aTP_Lamm_tt: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__tu____,type,
aTP_Lamm_tu: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__tv____,type,
aTP_Lamm_tv: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__tw____,type,
aTP_Lamm_tw: fun_nat_real > fun_real_real ).
tff(sy_c_ATP_058Lamm__tx____,type,
aTP_Lamm_tx: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__ty____,type,
aTP_Lamm_ty: fun_re4238438383274954274t_real > fun_real_real ).
tff(sy_c_ATP_058Lamm__tz____,type,
aTP_Lamm_tz: fun_real_real ).
tff(sy_c_ATP_058Lamm__ua____,type,
aTP_Lamm_ua: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__ub____,type,
aTP_Lamm_ub: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__uc____,type,
aTP_Lamm_uc: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__ud____,type,
aTP_Lamm_ud: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ue____,type,
aTP_Lamm_ue: fun_nat_real ).
tff(sy_c_ATP_058Lamm__uf____,type,
aTP_Lamm_uf: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ug____,type,
aTP_Lamm_ug: real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__uh____,type,
aTP_Lamm_uh: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ui____,type,
aTP_Lamm_ui: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__uj____,type,
aTP_Lamm_uj: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__uk____,type,
aTP_Lamm_uk: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ul____,type,
aTP_Lamm_ul: fun_nat_real > fun_fu6119738925743952495t_real ).
tff(sy_c_ATP_058Lamm__um____,type,
aTP_Lamm_um: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__un____,type,
aTP_Lamm_un: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__uo____,type,
aTP_Lamm_uo: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__up____,type,
aTP_Lamm_up: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__uq____,type,
aTP_Lamm_uq: nat > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__ur____,type,
aTP_Lamm_ur: fun_nat_o > fun_nat_o ).
tff(sy_c_ATP_058Lamm__us____,type,
aTP_Lamm_us: fun_nat_o > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__ut____,type,
aTP_Lamm_ut: real > fun_real_fun_real_o ).
tff(sy_c_ATP_058Lamm__uu____,type,
aTP_Lamm_uu: nat > fun_li6914387844523535256_nat_o ).
tff(sy_c_ATP_058Lamm__uv____,type,
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set_or1266510415728281911st_int: ( int * int ) > set_int ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
set_or7049704709247886629st_num: ( num * num ) > set_num ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
set_or633870826150836451st_rat: ( rat * rat ) > set_rat ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: ( real * real ) > set_real ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: ( int * int ) > set_int ).
tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: fun_nat_set_nat ).
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: ( $o * $o * $o * $o * $o * $o * $o * $o ) > 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_Pr8877776288780692068_nat_o ).
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_Pr8877776288780692068_nat_o ).
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_VE2962686792283380846VEBT_o ).
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_Pr8877776288780692068_nat_o ).
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_Pr8877776288780692068_nat_o ).
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_VE2962686792283380846VEBT_o ).
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_VE2962686792283380846VEBT_o ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
vEBT_T_p_r_e_d: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
vEBT_T_p_r_e_d2: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
vEBT_T_p_r_e_d_rel: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
vEBT_T_p_r_e_d_rel2: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
vEBT_T_s_u_c_c: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
vEBT_T_s_u_c_c2: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
vEBT_T_s_u_c_c_rel: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
vEBT_T_s_u_c_c_rel2: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: ( $o * $o ) > 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: vEBT_VEBT > fun_nat_o ).
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_o ).
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_o ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > fun_nat_o ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > fun_nat_o ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).
tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > fun_nat_o ).
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_o ).
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_VE2962686792283380846VEBT_o ).
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_Pr8877776288780692068_nat_o ).
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_o ).
tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).
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_o ).
tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).
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_op5794009454290973322_nat_o ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: set_nat > fun_nat_o ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: set_nat > fun_nat_o ).
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 ).
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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_fChoice_001t__Real__Oreal,type,
fChoice_real: fun_real_o > real ).
tff(sy_c_member_001_Eo,type,
member_o: ( $o * set_o ) > $o ).
tff(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > fun_set_complex_o ).
tff(sy_c_member_001t__Int__Oint,type,
member_int: int > fun_set_int_o ).
tff(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: ( list_o * set_list_o ) > $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_o ).
tff(sy_c_member_001t__Num__Onum,type,
member_num: num > fun_set_num_o ).
tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
member1379723562493234055eger_o: ( produc6271795597528267376eger_o * set_Pr448751882837621926eger_o ) > $o ).
tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).
tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).
tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).
tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $o ).
tff(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > fun_set_rat_o ).
tff(sy_c_member_001t__Real__Oreal,type,
member_real: real > fun_set_real_o ).
tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: ( set_nat * set_set_nat ) > $o ).
tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > fun_set_VEBT_VEBT_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 (10771)
tff(fact_0_degprop,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg) ).
% degprop
tff(fact_1__C5_Ohyps_C_I7_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).
% "5.hyps"(7)
tff(fact_2_max__in__set__def,axiom,
! [Xs: set_nat,Xb: nat] :
( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xb)
<=> ( aa_set_nat_o(member_nat(Xb),Xs)
& ! [X: nat] :
( aa_set_nat_o(member_nat(X),Xs)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xb) ) ) ) ).
% max_in_set_def
tff(fact_3_min__in__set__def,axiom,
! [Xs: set_nat,Xb: nat] :
( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xb)
<=> ( aa_set_nat_o(member_nat(Xb),Xs)
& ! [X: nat] :
( aa_set_nat_o(member_nat(X),Xs)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X) ) ) ) ).
% min_in_set_def
tff(fact_4_height__compose__summary,axiom,
! [Summarya: vEBT_VEBT,Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(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,Summarya))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Info,Dega,TreeLista,Summarya))) ).
% height_compose_summary
tff(fact_5__C5_OIH_C_I2_J,axiom,
! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(summary),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,summary))) ).
% "5.IH"(2)
tff(fact_6__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_7_nat__add__left__cancel__le,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% nat_add_left_cancel_le
tff(fact_8_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% add_le_cancel_left
tff(fact_9_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% add_le_cancel_left
tff(fact_10_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_cancel_left
tff(fact_11_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_cancel_left
tff(fact_12_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% add_le_cancel_right
tff(fact_13_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% add_le_cancel_right
tff(fact_14_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_cancel_right
tff(fact_15_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_cancel_right
tff(fact_16_height__compose__child,axiom,
! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Dega: nat,Summarya: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(aa_nat_fun_nat_o(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,Ta))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Info,Dega,TreeLista,Summarya))) ) ).
% height_compose_child
tff(fact_17_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_18_option_Oinject,axiom,
! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
<=> ( X2 = Y2 ) ) ).
% option.inject
tff(fact_19_option_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( some_nat(X2) = some_nat(Y2) )
<=> ( X2 = Y2 ) ) ).
% option.inject
tff(fact_20_option_Oinject,axiom,
! [X2: num,Y2: num] :
( ( some_num(X2) = some_num(Y2) )
<=> ( X2 = Y2 ) ) ).
% option.inject
tff(fact_21_prod_Oinject,axiom,
! [X1: code_integer,X2: $o,Y1: code_integer,Y2: $o] :
( ( produc6677183202524767010eger_o(X1,(X2)) = produc6677183202524767010eger_o(Y1,(Y2)) )
<=> ( ( X1 = Y1 )
& ( (X2)
<=> (Y2) ) ) ) ).
% prod.inject
tff(fact_22_prod_Oinject,axiom,
! [X1: num,X2: num,Y1: num,Y2: num] :
( ( product_Pair_num_num(X1,X2) = product_Pair_num_num(Y1,Y2) )
<=> ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
tff(fact_23_prod_Oinject,axiom,
! [X1: nat,X2: num,Y1: nat,Y2: num] :
( ( product_Pair_nat_num(X1,X2) = product_Pair_nat_num(Y1,Y2) )
<=> ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
tff(fact_24_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y2) )
<=> ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
tff(fact_25_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
<=> ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
tff(fact_26_old_Oprod_Oinject,axiom,
! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A2,(B2)) )
<=> ( ( A = A2 )
& ( (B)
<=> (B2) ) ) ) ).
% old.prod.inject
tff(fact_27_old_Oprod_Oinject,axiom,
! [A: num,B: num,A2: num,B2: num] :
( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_28_old_Oprod_Oinject,axiom,
! [A: nat,B: num,A2: nat,B2: num] :
( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_29_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_30_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_31_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_32_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_33_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_34_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_35_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_36_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_37_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_38_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_39_lesseq__shift,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
<=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xb)),some_nat(Y)) ) ).
% lesseq_shift
tff(fact_40__C5_Ohyps_C_I3_J,axiom,
m = aa_nat_nat(suc,na) ).
% "5.hyps"(3)
tff(fact_41__C5_Ohyps_C_I1_J,axiom,
aa_nat_o(vEBT_invar_vebt(summary),m) ).
% "5.hyps"(1)
tff(fact_42__C5_Ohyps_C_I6_J,axiom,
( ( mi = ma )
=> ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(treeList))
=> ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) ) ).
% "5.hyps"(6)
tff(fact_43__C5_OIH_C_I1_J,axiom,
! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(treeList))
=> ( aa_nat_o(vEBT_invar_vebt(X3),na)
& ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(X3),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X3))) ) ) ).
% "5.IH"(1)
tff(fact_44_one__reorient,axiom,
! [Xb: complex] :
( ( one_one_complex = Xb )
<=> ( Xb = one_one_complex ) ) ).
% one_reorient
tff(fact_45_one__reorient,axiom,
! [Xb: real] :
( ( one_one_real = Xb )
<=> ( Xb = one_one_real ) ) ).
% one_reorient
tff(fact_46_one__reorient,axiom,
! [Xb: rat] :
( ( one_one_rat = Xb )
<=> ( Xb = one_one_rat ) ) ).
% one_reorient
tff(fact_47_one__reorient,axiom,
! [Xb: nat] :
( ( one_one_nat = Xb )
<=> ( Xb = one_one_nat ) ) ).
% one_reorient
tff(fact_48_one__reorient,axiom,
! [Xb: int] :
( ( one_one_int = Xb )
<=> ( Xb = one_one_int ) ) ).
% one_reorient
tff(fact_49_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_50_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_51_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_52_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_53_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_54_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_55_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_56_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_57_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_58_add_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).
% add.left_commute
tff(fact_59_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_60_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_61_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_62_add_Ocommute,axiom,
! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).
% add.commute
tff(fact_63_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_64_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_65_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_66_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_67_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_68_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_69_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_70_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_71_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_72_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).
% add.assoc
tff(fact_73_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_74_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_75_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B3) = 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_76_group__cancel_Oadd2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3 = aa_rat_rat(plus_plus_rat(K),B) )
=> ( aa_rat_rat(plus_plus_rat(A),B3) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).
% group_cancel.add2
tff(fact_77_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3 = 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),B3) = 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_78_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3 = 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),B3) = 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_79_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A3),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_80_group__cancel_Oadd1,axiom,
! [A3: rat,K: rat,A: rat,B: rat] :
( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
=> ( aa_rat_rat(plus_plus_rat(A3),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).
% group_cancel.add1
tff(fact_81_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),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_82_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),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_83_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_84_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_85_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_86_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_87_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_88_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).
% ab_semigroup_add_class.add_ac(1)
tff(fact_89_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_90_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_91_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
( member8440522571783428010at_nat(A,collec3392354462482085612at_nat(P))
<=> aa_Pro6150637604445964947_nat_o(P,A) ) ).
% mem_Collect_eq
tff(fact_92_mem__Collect__eq,axiom,
! [A: real,P: fun_real_o] :
( aa_set_real_o(member_real(A),collect_real(P))
<=> aa_real_o(P,A) ) ).
% mem_Collect_eq
tff(fact_93_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
( member5262025264175285858nt_int(A,collec213857154873943460nt_int(P))
<=> aa_Pro4592068669816069595_int_o(P,A) ) ).
% mem_Collect_eq
tff(fact_94_mem__Collect__eq,axiom,
! [A: complex,P: fun_complex_o] :
( aa_set_complex_o(member_complex(A),collect_complex(P))
<=> aa_complex_o(P,A) ) ).
% mem_Collect_eq
tff(fact_95_mem__Collect__eq,axiom,
! [A: list_nat,P: fun_list_nat_o] :
( member_list_nat(A,collect_list_nat(P))
<=> aa_list_nat_o(P,A) ) ).
% mem_Collect_eq
tff(fact_96_mem__Collect__eq,axiom,
! [A: nat,P: fun_nat_o] :
( aa_set_nat_o(member_nat(A),collect_nat(P))
<=> aa_nat_o(P,A) ) ).
% mem_Collect_eq
tff(fact_97_mem__Collect__eq,axiom,
! [A: int,P: fun_int_o] :
( aa_set_int_o(member_int(A),collect_int(P))
<=> aa_int_o(P,A) ) ).
% mem_Collect_eq
tff(fact_98_Collect__mem__eq,axiom,
! [A3: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A3)) = A3 ).
% Collect_mem_eq
tff(fact_99_Collect__mem__eq,axiom,
! [A3: set_real] : collect_real(aTP_Lamm_aa(A3)) = A3 ).
% Collect_mem_eq
tff(fact_100_Collect__mem__eq,axiom,
! [A3: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ab(A3)) = A3 ).
% Collect_mem_eq
tff(fact_101_Collect__mem__eq,axiom,
! [A3: set_complex] : collect_complex(aTP_Lamm_ac(A3)) = A3 ).
% Collect_mem_eq
tff(fact_102_Collect__mem__eq,axiom,
! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ad(A3)) = A3 ).
% Collect_mem_eq
tff(fact_103_Collect__mem__eq,axiom,
! [A3: set_nat] : collect_nat(aTP_Lamm_ae(A3)) = A3 ).
% Collect_mem_eq
tff(fact_104_Collect__mem__eq,axiom,
! [A3: set_int] : collect_int(aTP_Lamm_af(A3)) = A3 ).
% Collect_mem_eq
tff(fact_105_Collect__cong,axiom,
! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
( ! [X4: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(P,X4)
<=> aa_Pro4592068669816069595_int_o(Q,X4) )
=> ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).
% Collect_cong
tff(fact_106_Collect__cong,axiom,
! [P: fun_complex_o,Q: fun_complex_o] :
( ! [X4: complex] :
( aa_complex_o(P,X4)
<=> aa_complex_o(Q,X4) )
=> ( collect_complex(P) = collect_complex(Q) ) ) ).
% Collect_cong
tff(fact_107_Collect__cong,axiom,
! [P: fun_list_nat_o,Q: fun_list_nat_o] :
( ! [X4: list_nat] :
( aa_list_nat_o(P,X4)
<=> aa_list_nat_o(Q,X4) )
=> ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).
% Collect_cong
tff(fact_108_Collect__cong,axiom,
! [P: fun_nat_o,Q: fun_nat_o] :
( ! [X4: nat] :
( aa_nat_o(P,X4)
<=> aa_nat_o(Q,X4) )
=> ( collect_nat(P) = collect_nat(Q) ) ) ).
% Collect_cong
tff(fact_109_Collect__cong,axiom,
! [P: fun_int_o,Q: fun_int_o] :
( ! [X4: int] :
( aa_int_o(P,X4)
<=> aa_int_o(Q,X4) )
=> ( collect_int(P) = collect_int(Q) ) ) ).
% Collect_cong
tff(fact_110_Pair__inject,axiom,
! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A2,(B2)) )
=> ~ ( ( A = A2 )
=> ( (B)
<=> ~ (B2) ) ) ) ).
% Pair_inject
tff(fact_111_Pair__inject,axiom,
! [A: num,B: num,A2: num,B2: num] :
( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_112_Pair__inject,axiom,
! [A: nat,B: num,A2: nat,B2: num] :
( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_113_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_114_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_115_prod__cases,axiom,
! [P: fun_Pr3290071492984375169er_o_o,P2: produc6271795597528267376eger_o] :
( ! [A4: code_integer,B4: $o] : aa_Pro5286387174248033258er_o_o(P,produc6677183202524767010eger_o(A4,(B4)))
=> aa_Pro5286387174248033258er_o_o(P,P2) ) ).
% prod_cases
tff(fact_116_prod__cases,axiom,
! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
( ! [A4: num,B4: num] : aa_Pro7685007914727650815_num_o(P,product_Pair_num_num(A4,B4))
=> aa_Pro7685007914727650815_num_o(P,P2) ) ).
% prod_cases
tff(fact_117_prod__cases,axiom,
! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
( ! [A4: nat,B4: num] : aa_Pro6908818167415475913_num_o(P,product_Pair_nat_num(A4,B4))
=> aa_Pro6908818167415475913_num_o(P,P2) ) ).
% prod_cases
tff(fact_118_prod__cases,axiom,
! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4))
=> aa_Pro6150637604445964947_nat_o(P,P2) ) ).
% prod_cases
tff(fact_119_prod__cases,axiom,
! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
( ! [A4: int,B4: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4))
=> aa_Pro4592068669816069595_int_o(P,P2) ) ).
% prod_cases
tff(fact_120_surj__pair,axiom,
! [P2: produc6271795597528267376eger_o] :
? [X4: code_integer,Y3: $o] : P2 = produc6677183202524767010eger_o(X4,(Y3)) ).
% surj_pair
tff(fact_121_surj__pair,axiom,
! [P2: product_prod_num_num] :
? [X4: num,Y3: num] : P2 = product_Pair_num_num(X4,Y3) ).
% surj_pair
tff(fact_122_surj__pair,axiom,
! [P2: product_prod_nat_num] :
? [X4: nat,Y3: num] : P2 = product_Pair_nat_num(X4,Y3) ).
% surj_pair
tff(fact_123_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X4: nat,Y3: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) ).
% surj_pair
tff(fact_124_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X4: int,Y3: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) ).
% surj_pair
tff(fact_125_old_Oprod_Oexhaust,axiom,
! [Y: produc6271795597528267376eger_o] :
~ ! [A4: code_integer,B4: $o] : Y != produc6677183202524767010eger_o(A4,(B4)) ).
% old.prod.exhaust
tff(fact_126_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_127_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_128_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_129_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_130_Nat_Oex__has__greatest__nat,axiom,
! [P: fun_nat_o,K: nat,B: nat] :
( aa_nat_o(P,K)
=> ( ! [Y3: nat] :
( aa_nat_o(P,Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
=> ? [X4: nat] :
( aa_nat_o(P,X4)
& ! [Y4: nat] :
( aa_nat_o(P,Y4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X4) ) ) ) ) ).
% Nat.ex_has_greatest_nat
tff(fact_131_nat__le__linear,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).
% nat_le_linear
tff(fact_132_le__antisym,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( Mb = Nb ) ) ) ).
% le_antisym
tff(fact_133_eq__imp__le,axiom,
! [Mb: nat,Nb: nat] :
( ( Mb = Nb )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% eq_imp_le
tff(fact_134_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).
% le_trans
tff(fact_135_le__refl,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).
% le_refl
tff(fact_136_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% add_le_imp_le_right
tff(fact_137_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_imp_le_right
tff(fact_138_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_imp_le_right
tff(fact_139_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% add_le_imp_le_right
tff(fact_140_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% add_le_imp_le_left
tff(fact_141_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_imp_le_left
tff(fact_142_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_imp_le_left
tff(fact_143_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% add_le_imp_le_left
tff(fact_144_le__iff__add,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ? [C2: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).
% le_iff_add
tff(fact_145_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).
% add_right_mono
tff(fact_146_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_147_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_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_148_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_149_less__eqE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).
% less_eqE
tff(fact_150_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).
% add_left_mono
tff(fact_151_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_152_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_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_153_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_154_add__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).
% add_mono
tff(fact_155_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).
% add_mono
tff(fact_156_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
=> aa_int_o(aa_int_fun_int_o(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_157_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).
% add_mono
tff(fact_158_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_159_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_160_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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_161_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_162_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_163_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_164_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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_165_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
& ( K = L ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
& ( K = L ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_168_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
& ( K = L ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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_169_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
& ( K = L ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_170_nat__le__iff__add,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
<=> ? [K2: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2) ) ).
% nat_le_iff_add
tff(fact_171_trans__le__add2,axiom,
! [I: nat,J: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).
% trans_le_add2
tff(fact_172_trans__le__add1,axiom,
! [I: nat,J: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).
% trans_le_add1
tff(fact_173_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).
% add_le_mono1
tff(fact_174_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).
% add_le_mono
tff(fact_175_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
=> ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).
% le_Suc_ex
tff(fact_176_add__leD2,axiom,
! [Mb: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).
% add_leD2
tff(fact_177_add__leD1,axiom,
! [Mb: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% add_leD1
tff(fact_178_le__add2,axiom,
! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).
% le_add2
tff(fact_179_le__add1,axiom,
! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).
% le_add1
tff(fact_180_add__leE,axiom,
! [Mb: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).
% add_leE
tff(fact_181_insert__simp__mima,axiom,
! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( ( ( Xb = Mia )
| ( Xb = Maa ) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).
% insert_simp_mima
tff(fact_182_one__add__one,axiom,
aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_183_one__add__one,axiom,
plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_184_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_185_one__add__one,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_186_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_187_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_188_one__plus__numeral,axiom,
! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% one_plus_numeral
tff(fact_189_one__plus__numeral,axiom,
! [Nb: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Nb)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% one_plus_numeral
tff(fact_190_one__plus__numeral,axiom,
! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% one_plus_numeral
tff(fact_191_one__plus__numeral,axiom,
! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% one_plus_numeral
tff(fact_192_one__plus__numeral,axiom,
! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% one_plus_numeral
tff(fact_193_one__plus__numeral,axiom,
! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% one_plus_numeral
tff(fact_194_numeral__plus__one,axiom,
! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% numeral_plus_one
tff(fact_195_numeral__plus__one,axiom,
! [Nb: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Nb),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% numeral_plus_one
tff(fact_196_numeral__plus__one,axiom,
! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% numeral_plus_one
tff(fact_197_numeral__plus__one,axiom,
! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% numeral_plus_one
tff(fact_198_numeral__plus__one,axiom,
! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% numeral_plus_one
tff(fact_199_numeral__plus__one,axiom,
! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% numeral_plus_one
tff(fact_200_numeral__le__one__iff,axiom,
! [Nb: num] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Nb)),one_on7984719198319812577d_enat)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).
% numeral_le_one_iff
tff(fact_201_numeral__le__one__iff,axiom,
! [Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).
% numeral_le_one_iff
tff(fact_202_numeral__le__one__iff,axiom,
! [Nb: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).
% numeral_le_one_iff
tff(fact_203_numeral__le__one__iff,axiom,
! [Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).
% numeral_le_one_iff
tff(fact_204_numeral__le__one__iff,axiom,
! [Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).
% numeral_le_one_iff
tff(fact_205_one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
<=> ( one = Nb ) ) ).
% one_eq_numeral_iff
tff(fact_206_one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( one_on7984719198319812577d_enat = numera1916890842035813515d_enat(Nb) )
<=> ( one = Nb ) ) ).
% one_eq_numeral_iff
tff(fact_207_one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( one_one_complex = numera6690914467698888265omplex(Nb) )
<=> ( one = Nb ) ) ).
% one_eq_numeral_iff
tff(fact_208_one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
<=> ( one = Nb ) ) ).
% one_eq_numeral_iff
tff(fact_209_one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
<=> ( one = Nb ) ) ).
% one_eq_numeral_iff
tff(fact_210_one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
<=> ( one = Nb ) ) ).
% one_eq_numeral_iff
tff(fact_211_numeral__eq__one__iff,axiom,
! [Nb: num] :
( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
<=> ( Nb = one ) ) ).
% numeral_eq_one_iff
tff(fact_212_numeral__eq__one__iff,axiom,
! [Nb: num] :
( ( numera1916890842035813515d_enat(Nb) = one_on7984719198319812577d_enat )
<=> ( Nb = one ) ) ).
% numeral_eq_one_iff
tff(fact_213_numeral__eq__one__iff,axiom,
! [Nb: num] :
( ( numera6690914467698888265omplex(Nb) = one_one_complex )
<=> ( Nb = one ) ) ).
% numeral_eq_one_iff
tff(fact_214_numeral__eq__one__iff,axiom,
! [Nb: num] :
( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
<=> ( Nb = one ) ) ).
% numeral_eq_one_iff
tff(fact_215_numeral__eq__one__iff,axiom,
! [Nb: num] :
( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
<=> ( Nb = one ) ) ).
% numeral_eq_one_iff
tff(fact_216_numeral__eq__one__iff,axiom,
! [Nb: num] :
( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
<=> ( Nb = one ) ) ).
% numeral_eq_one_iff
tff(fact_217_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_218_semiring__norm_I85_J,axiom,
! [Mb: num] : aa_num_num(bit0,Mb) != one ).
% semiring_norm(85)
tff(fact_219_semiring__norm_I83_J,axiom,
! [Nb: num] : one != aa_num_num(bit0,Nb) ).
% semiring_norm(83)
tff(fact_220_semiring__norm_I69_J,axiom,
! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).
% semiring_norm(69)
tff(fact_221_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_222_even__odd__cases,axiom,
! [Xb: nat] :
( ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
=> ~ ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).
% even_odd_cases
tff(fact_223_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
=> ( Dega = Nb ) ) ).
% deg_deg_n
tff(fact_224_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
=> ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S) ) ).
% deg_SUcn_Node
tff(fact_225_numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( numera1916890842035813515d_enat(Mb) = numera1916890842035813515d_enat(Nb) )
<=> ( Mb = Nb ) ) ).
% numeral_eq_iff
tff(fact_226_numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
<=> ( Mb = Nb ) ) ).
% numeral_eq_iff
tff(fact_227_numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_num_real(numeral_numeral_real,Mb) = aa_num_real(numeral_numeral_real,Nb) )
<=> ( Mb = Nb ) ) ).
% numeral_eq_iff
tff(fact_228_numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
<=> ( Mb = Nb ) ) ).
% numeral_eq_iff
tff(fact_229_numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
<=> ( Mb = Nb ) ) ).
% numeral_eq_iff
tff(fact_230_semiring__norm_I87_J,axiom,
! [Mb: num,Nb: num] :
( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
<=> ( Mb = Nb ) ) ).
% semiring_norm(87)
tff(fact_231_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
<=> ( X2 = Y2 ) ) ).
% nat.inject
tff(fact_232_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_233_mi__eq__ma__no__ch,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
=> ( ( Mia = Maa )
=> ( ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
=> ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) )
& ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).
% mi_eq_ma_no_ch
tff(fact_234_Suc__le__mono,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).
% Suc_le_mono
tff(fact_235_add__Suc__right,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).
% add_Suc_right
tff(fact_236_semiring__norm_I6_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% semiring_norm(6)
tff(fact_237_semiring__norm_I71_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% semiring_norm(71)
tff(fact_238_semiring__norm_I68_J,axiom,
! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).
% semiring_norm(68)
tff(fact_239_numeral__plus__numeral,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% numeral_plus_numeral
tff(fact_240_numeral__plus__numeral,axiom,
! [Mb: num,Nb: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% numeral_plus_numeral
tff(fact_241_numeral__plus__numeral,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% numeral_plus_numeral
tff(fact_242_numeral__plus__numeral,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% numeral_plus_numeral
tff(fact_243_numeral__plus__numeral,axiom,
! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% numeral_plus_numeral
tff(fact_244_numeral__plus__numeral,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% numeral_plus_numeral
tff(fact_245_add__numeral__left,axiom,
! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).
% add_numeral_left
tff(fact_246_add__numeral__left,axiom,
! [V: num,W: num,Z: extended_enat] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W),Z)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).
% add_numeral_left
tff(fact_247_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_248_add__numeral__left,axiom,
! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).
% add_numeral_left
tff(fact_249_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_250_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_251_numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% numeral_le_iff
tff(fact_252_numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% numeral_le_iff
tff(fact_253_numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% numeral_le_iff
tff(fact_254_numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% numeral_le_iff
tff(fact_255_numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% numeral_le_iff
tff(fact_256_Suc__numeral,axiom,
! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% Suc_numeral
tff(fact_257_add__2__eq__Suc_H,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).
% add_2_eq_Suc'
tff(fact_258_add__2__eq__Suc,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).
% add_2_eq_Suc
tff(fact_259_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_260_Suc__inject,axiom,
! [Xb: nat,Y: nat] :
( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
=> ( Xb = Y ) ) ).
% Suc_inject
tff(fact_261_n__not__Suc__n,axiom,
! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).
% n_not_Suc_n
tff(fact_262_Suc__leD,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% Suc_leD
tff(fact_263_le__SucE,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).
% le_SucE
tff(fact_264_le__SucI,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).
% le_SucI
tff(fact_265_Suc__le__D,axiom,
! [Nb: nat,M: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M)
=> ? [M2: nat] : M = aa_nat_nat(suc,M2) ) ).
% Suc_le_D
tff(fact_266_le__Suc__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
| ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).
% le_Suc_eq
tff(fact_267_Suc__n__not__le__n,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).
% Suc_n_not_le_n
tff(fact_268_not__less__eq__eq,axiom,
! [Mb: nat,Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).
% not_less_eq_eq
tff(fact_269_full__nat__induct,axiom,
! [P: fun_nat_o,Nb: nat] :
( ! [N: nat] :
( ! [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M3)),N)
=> aa_nat_o(P,M3) )
=> aa_nat_o(P,N) )
=> aa_nat_o(P,Nb) ) ).
% full_nat_induct
tff(fact_270_nat__induct__at__least,axiom,
! [Mb: nat,Nb: nat,P: fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_o(P,Mb)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
=> ( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(suc,N)) ) )
=> aa_nat_o(P,Nb) ) ) ) ).
% nat_induct_at_least
tff(fact_271_transitive__stepwise__le,axiom,
! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X4),X4)
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(R,X4),Y3)
=> ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
=> aa_nat_o(aa_nat_fun_nat_o(R,X4),Z2) ) )
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
=> aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).
% transitive_stepwise_le
tff(fact_272_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
=> ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).
% nat_arith.suc1
tff(fact_273_add__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).
% add_Suc
tff(fact_274_add__Suc__shift,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).
% add_Suc_shift
tff(fact_275_Suc__nat__number__of__add,axiom,
! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = 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))),Nb) ).
% Suc_nat_number_of_add
tff(fact_276_lift__Suc__mono__le,axiom,
! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N2)) ) ) ).
% lift_Suc_mono_le
tff(fact_277_lift__Suc__mono__le,axiom,
! [F: fun_nat_rat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).
% lift_Suc_mono_le
tff(fact_278_lift__Suc__mono__le,axiom,
! [F: fun_nat_num,Nb: nat,N2: nat] :
( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).
% lift_Suc_mono_le
tff(fact_279_lift__Suc__mono__le,axiom,
! [F: fun_nat_nat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).
% lift_Suc_mono_le
tff(fact_280_lift__Suc__mono__le,axiom,
! [F: fun_nat_int,Nb: nat,N2: nat] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).
% lift_Suc_mono_le
tff(fact_281_lift__Suc__mono__le,axiom,
! [F: fun_nat_real,Nb: nat,N2: nat] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).
% lift_Suc_mono_le
tff(fact_282_lift__Suc__antimono__le,axiom,
! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,Nb)) ) ) ).
% lift_Suc_antimono_le
tff(fact_283_lift__Suc__antimono__le,axiom,
! [F: fun_nat_rat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Nb)) ) ) ).
% lift_Suc_antimono_le
tff(fact_284_lift__Suc__antimono__le,axiom,
! [F: fun_nat_num,Nb: nat,N2: nat] :
( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Nb)) ) ) ).
% lift_Suc_antimono_le
tff(fact_285_lift__Suc__antimono__le,axiom,
! [F: fun_nat_nat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Nb)) ) ) ).
% lift_Suc_antimono_le
tff(fact_286_lift__Suc__antimono__le,axiom,
! [F: fun_nat_int,Nb: nat,N2: nat] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Nb)) ) ) ).
% lift_Suc_antimono_le
tff(fact_287_lift__Suc__antimono__le,axiom,
! [F: fun_nat_real,Nb: nat,N2: nat] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,Nb)) ) ) ).
% lift_Suc_antimono_le
tff(fact_288_Suc__eq__plus1,axiom,
! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).
% Suc_eq_plus1
tff(fact_289_plus__1__eq__Suc,axiom,
aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).
% plus_1_eq_Suc
tff(fact_290_Suc__eq__plus1__left,axiom,
! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).
% Suc_eq_plus1_left
tff(fact_291_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_292_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).
% is_num_normalize(1)
tff(fact_293_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_294_add__One__commute,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).
% add_One_commute
tff(fact_295_le__num__One__iff,axiom,
! [Xb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
<=> ( Xb = one ) ) ).
% le_num_One_iff
tff(fact_296_height__node,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(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(Mia),Maa)),Dega,TreeLista,Summarya))) ) ).
% height_node
tff(fact_297_le__numeral__extra_I4_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).
% le_numeral_extra(4)
tff(fact_298_le__numeral__extra_I4_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).
% le_numeral_extra(4)
tff(fact_299_le__numeral__extra_I4_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).
% le_numeral_extra(4)
tff(fact_300_le__numeral__extra_I4_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).
% le_numeral_extra(4)
tff(fact_301_one__le__numeral,axiom,
! [Nb: num] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Nb)) ).
% one_le_numeral
tff(fact_302_one__le__numeral,axiom,
! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).
% one_le_numeral
tff(fact_303_one__le__numeral,axiom,
! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).
% one_le_numeral
tff(fact_304_one__le__numeral,axiom,
! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).
% one_le_numeral
tff(fact_305_one__le__numeral,axiom,
! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).
% one_le_numeral
tff(fact_306_one__plus__numeral__commute,axiom,
! [Xb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).
% one_plus_numeral_commute
tff(fact_307_one__plus__numeral__commute,axiom,
! [Xb: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xb)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xb),one_on7984719198319812577d_enat) ).
% one_plus_numeral_commute
tff(fact_308_one__plus__numeral__commute,axiom,
! [Xb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).
% one_plus_numeral_commute
tff(fact_309_one__plus__numeral__commute,axiom,
! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).
% one_plus_numeral_commute
tff(fact_310_one__plus__numeral__commute,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).
% one_plus_numeral_commute
tff(fact_311_one__plus__numeral__commute,axiom,
! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).
% one_plus_numeral_commute
tff(fact_312_numeral__Bit0,axiom,
! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).
% numeral_Bit0
tff(fact_313_numeral__Bit0,axiom,
! [Nb: num] : numera1916890842035813515d_enat(aa_num_num(bit0,Nb)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Nb),numera1916890842035813515d_enat(Nb)) ).
% numeral_Bit0
tff(fact_314_numeral__Bit0,axiom,
! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).
% numeral_Bit0
tff(fact_315_numeral__Bit0,axiom,
! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).
% numeral_Bit0
tff(fact_316_numeral__Bit0,axiom,
! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).
% numeral_Bit0
tff(fact_317_numeral__Bit0,axiom,
! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).
% numeral_Bit0
tff(fact_318_numeral__One,axiom,
aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).
% numeral_One
tff(fact_319_numeral__One,axiom,
numera1916890842035813515d_enat(one) = one_on7984719198319812577d_enat ).
% numeral_One
tff(fact_320_numeral__One,axiom,
numera6690914467698888265omplex(one) = one_one_complex ).
% numeral_One
tff(fact_321_numeral__One,axiom,
aa_num_real(numeral_numeral_real,one) = one_one_real ).
% numeral_One
tff(fact_322_numeral__One,axiom,
aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).
% numeral_One
tff(fact_323_numeral__One,axiom,
aa_num_int(numeral_numeral_int,one) = one_one_int ).
% numeral_One
tff(fact_324_numerals_I1_J,axiom,
aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).
% numerals(1)
tff(fact_325_enat__ord__number_I1_J,axiom,
! [Mb: num,Nb: num] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).
% enat_ord_number(1)
tff(fact_326_add__shift,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = Z )
<=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).
% add_shift
tff(fact_327_maxbmo,axiom,
! [Ta: vEBT_VEBT,Xb: nat] :
( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
=> aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb) ) ).
% maxbmo
tff(fact_328_dbl__simps_I3_J,axiom,
aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_329_dbl__simps_I3_J,axiom,
aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_330_dbl__simps_I3_J,axiom,
aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_331_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_332__C5_Ohyps_C_I8_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(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_333_member__bound__height_H,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).
% member_bound_height'
tff(fact_334__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_335_valid__member__both__member__options,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).
% valid_member_both_member_options
tff(fact_336_both__member__options__equiv__member,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
<=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).
% both_member_options_equiv_member
tff(fact_337_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
<=> ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
tff(fact_338_order__refl,axiom,
! [Xb: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Xb),Xb) ).
% order_refl
tff(fact_339_order__refl,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).
% order_refl
tff(fact_340_order__refl,axiom,
! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).
% order_refl
tff(fact_341_order__refl,axiom,
! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).
% order_refl
tff(fact_342_order__refl,axiom,
! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).
% order_refl
tff(fact_343_order__refl,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Xb) ).
% order_refl
tff(fact_344_power__shift,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y) = Z )
<=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).
% power_shift
tff(fact_345_dual__order_Orefl,axiom,
! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).
% dual_order.refl
tff(fact_346_dual__order_Orefl,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).
% dual_order.refl
tff(fact_347_dual__order_Orefl,axiom,
! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).
% dual_order.refl
tff(fact_348_dual__order_Orefl,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).
% dual_order.refl
tff(fact_349_dual__order_Orefl,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).
% dual_order.refl
tff(fact_350_dual__order_Orefl,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),A) ).
% dual_order.refl
tff(fact_351_maxt__member,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
=> aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).
% maxt_member
tff(fact_352_maxt__corr__help,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
=> ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maxi) ) ) ) ).
% maxt_corr_help
tff(fact_353_member__bound,axiom,
! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
( aa_nat_o(vEBT_vebt_member(Tree),Xb)
=> ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).
% member_bound
tff(fact_354_numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% numeral_less_iff
tff(fact_355_numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% numeral_less_iff
tff(fact_356_numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% numeral_less_iff
tff(fact_357_numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% numeral_less_iff
tff(fact_358_numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% numeral_less_iff
tff(fact_359_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% add_less_cancel_right
tff(fact_360_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% add_less_cancel_right
tff(fact_361_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_cancel_right
tff(fact_362_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_cancel_right
tff(fact_363_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% add_less_cancel_left
tff(fact_364_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% add_less_cancel_left
tff(fact_365_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_cancel_left
tff(fact_366_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_cancel_left
tff(fact_367_Suc__less__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% Suc_less_eq
tff(fact_368_Suc__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).
% Suc_mono
tff(fact_369_lessI,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).
% lessI
tff(fact_370_nat__add__left__cancel__less,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% nat_add_left_cancel_less
tff(fact_371_valid__insert__both__member__options__add,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).
% valid_insert_both_member_options_add
tff(fact_372_valid__insert__both__member__options__pres,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(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))),Nb))
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
=> aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).
% valid_insert_both_member_options_pres
tff(fact_373_post__member__pre__member,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(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))),Nb))
=> ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
| ( Xb = Y ) ) ) ) ) ) ).
% post_member_pre_member
tff(fact_374_member__correct,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
<=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).
% member_correct
tff(fact_375_set__n__deg__not__0,axiom,
! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(vEBT_invar_vebt(X4),Nb) )
=> ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).
% set_n_deg_not_0
tff(fact_376_mi__ma__2__deg,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).
% mi_ma_2_deg
tff(fact_377_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_378_dbl__simps_I5_J,axiom,
! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).
% dbl_simps(5)
tff(fact_379_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_380_one__less__numeral__iff,axiom,
! [Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).
% one_less_numeral_iff
tff(fact_381_one__less__numeral__iff,axiom,
! [Nb: num] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).
% one_less_numeral_iff
tff(fact_382_one__less__numeral__iff,axiom,
! [Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).
% one_less_numeral_iff
tff(fact_383_one__less__numeral__iff,axiom,
! [Nb: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).
% one_less_numeral_iff
tff(fact_384_one__less__numeral__iff,axiom,
! [Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).
% one_less_numeral_iff
tff(fact_385_add__def,axiom,
vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).
% add_def
tff(fact_386__C5_Ohyps_C_I5_J,axiom,
! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I2)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I2) ) ) ).
% "5.hyps"(5)
tff(fact_387_verit__comp__simplify1_I1_J,axiom,
! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).
% verit_comp_simplify1(1)
tff(fact_388_verit__comp__simplify1_I1_J,axiom,
! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).
% verit_comp_simplify1(1)
tff(fact_389_verit__comp__simplify1_I1_J,axiom,
! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).
% verit_comp_simplify1(1)
tff(fact_390_verit__comp__simplify1_I1_J,axiom,
! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).
% verit_comp_simplify1(1)
tff(fact_391_verit__comp__simplify1_I1_J,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).
% verit_comp_simplify1(1)
tff(fact_392_lt__ex,axiom,
! [Xb: real] :
? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ).
% lt_ex
tff(fact_393_lt__ex,axiom,
! [Xb: rat] :
? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ).
% lt_ex
tff(fact_394_lt__ex,axiom,
! [Xb: int] :
? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ).
% lt_ex
tff(fact_395_gt__ex,axiom,
! [Xb: real] :
? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_13) ).
% gt_ex
tff(fact_396_gt__ex,axiom,
! [Xb: rat] :
? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_13) ).
% gt_ex
tff(fact_397_gt__ex,axiom,
! [Xb: nat] :
? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_13) ).
% gt_ex
tff(fact_398_gt__ex,axiom,
! [Xb: int] :
? [X_13: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_13) ).
% gt_ex
tff(fact_399_dense,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ? [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Y) ) ) ).
% dense
tff(fact_400_dense,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ? [Z2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z2)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Y) ) ) ).
% dense
tff(fact_401_less__imp__neq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( Xb != Y ) ) ).
% less_imp_neq
tff(fact_402_less__imp__neq,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( Xb != Y ) ) ).
% less_imp_neq
tff(fact_403_less__imp__neq,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( Xb != Y ) ) ).
% less_imp_neq
tff(fact_404_less__imp__neq,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( Xb != Y ) ) ).
% less_imp_neq
tff(fact_405_less__imp__neq,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( Xb != Y ) ) ).
% less_imp_neq
tff(fact_406_order_Oasym,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).
% order.asym
tff(fact_407_order_Oasym,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).
% order.asym
tff(fact_408_order_Oasym,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).
% order.asym
tff(fact_409_order_Oasym,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).
% order.asym
tff(fact_410_order_Oasym,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).
% order.asym
tff(fact_411_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_412_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_413_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_414_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_415_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_416_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( B = C )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_417_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( ( B = C )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_418_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( ( B = C )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_419_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( B = C )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_420_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( ( B = C )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_421_less__induct,axiom,
! [P: fun_nat_o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X4)
=> aa_nat_o(P,Y4) )
=> aa_nat_o(P,X4) )
=> aa_nat_o(P,A) ) ).
% less_induct
tff(fact_422_antisym__conv3,axiom,
! [Y: real,Xb: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv3
tff(fact_423_antisym__conv3,axiom,
! [Y: rat,Xb: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv3
tff(fact_424_antisym__conv3,axiom,
! [Y: num,Xb: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv3
tff(fact_425_antisym__conv3,axiom,
! [Y: nat,Xb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv3
tff(fact_426_antisym__conv3,axiom,
! [Y: int,Xb: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv3
tff(fact_427_linorder__cases,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( ( Xb != Y )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).
% linorder_cases
tff(fact_428_linorder__cases,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( ( Xb != Y )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).
% linorder_cases
tff(fact_429_linorder__cases,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( ( Xb != Y )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).
% linorder_cases
tff(fact_430_linorder__cases,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( ( Xb != Y )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).
% linorder_cases
tff(fact_431_linorder__cases,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( ( Xb != Y )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).
% linorder_cases
tff(fact_432_dual__order_Oasym,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% dual_order.asym
tff(fact_433_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% dual_order.asym
tff(fact_434_dual__order_Oasym,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).
% dual_order.asym
tff(fact_435_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% dual_order.asym
tff(fact_436_dual__order_Oasym,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% dual_order.asym
tff(fact_437_dual__order_Oirrefl,axiom,
! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).
% dual_order.irrefl
tff(fact_438_dual__order_Oirrefl,axiom,
! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).
% dual_order.irrefl
tff(fact_439_dual__order_Oirrefl,axiom,
! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).
% dual_order.irrefl
tff(fact_440_dual__order_Oirrefl,axiom,
! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).
% dual_order.irrefl
tff(fact_441_dual__order_Oirrefl,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).
% dual_order.irrefl
tff(fact_442_exists__least__iff,axiom,
! [P: fun_nat_o] :
( ? [X_12: nat] : aa_nat_o(P,X_12)
<=> ? [N3: nat] :
( aa_nat_o(P,N3)
& ! [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N3)
=> ~ aa_nat_o(P,M4) ) ) ) ).
% exists_least_iff
tff(fact_443_linorder__less__wlog,axiom,
! [P: fun_real_fun_real_o,A: real,B: real] :
( ! [A4: real,B4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B4)
=> aa_real_o(aa_real_fun_real_o(P,A4),B4) )
=> ( ! [A4: real] : aa_real_o(aa_real_fun_real_o(P,A4),A4)
=> ( ! [A4: real,B4: real] :
( aa_real_o(aa_real_fun_real_o(P,B4),A4)
=> aa_real_o(aa_real_fun_real_o(P,A4),B4) )
=> aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_444_linorder__less__wlog,axiom,
! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
( ! [A4: rat,B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4)
=> aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
=> ( ! [A4: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A4),A4)
=> ( ! [A4: rat,B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(P,B4),A4)
=> aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
=> aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_445_linorder__less__wlog,axiom,
! [P: fun_num_fun_num_o,A: num,B: num] :
( ! [A4: num,B4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B4)
=> aa_num_o(aa_num_fun_num_o(P,A4),B4) )
=> ( ! [A4: num] : aa_num_o(aa_num_fun_num_o(P,A4),A4)
=> ( ! [A4: num,B4: num] :
( aa_num_o(aa_num_fun_num_o(P,B4),A4)
=> aa_num_o(aa_num_fun_num_o(P,A4),B4) )
=> aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_446_linorder__less__wlog,axiom,
! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
=> aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
=> ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),A4)
=> ( ! [A4: nat,B4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,B4),A4)
=> aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
=> aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_447_linorder__less__wlog,axiom,
! [P: fun_int_fun_int_o,A: int,B: int] :
( ! [A4: int,B4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
=> aa_int_o(aa_int_fun_int_o(P,A4),B4) )
=> ( ! [A4: int] : aa_int_o(aa_int_fun_int_o(P,A4),A4)
=> ( ! [A4: int,B4: int] :
( aa_int_o(aa_int_fun_int_o(P,B4),A4)
=> aa_int_o(aa_int_fun_int_o(P,A4),B4) )
=> aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_448_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).
% order.strict_trans
tff(fact_449_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).
% order.strict_trans
tff(fact_450_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).
% order.strict_trans
tff(fact_451_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% order.strict_trans
tff(fact_452_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% order.strict_trans
tff(fact_453_not__less__iff__gr__or__eq,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
| ( Xb = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_454_not__less__iff__gr__or__eq,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
| ( Xb = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_455_not__less__iff__gr__or__eq,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
| ( Xb = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_456_not__less__iff__gr__or__eq,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
| ( Xb = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_457_not__less__iff__gr__or__eq,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
| ( Xb = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_458_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_459_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_460_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_461_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_462_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_463_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_464_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_465_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_466_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_467_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_468_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_469_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_470_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_471_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_472_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_473_linorder__neqE,axiom,
! [Xb: real,Y: real] :
( ( Xb != Y )
=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).
% linorder_neqE
tff(fact_474_linorder__neqE,axiom,
! [Xb: rat,Y: rat] :
( ( Xb != Y )
=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).
% linorder_neqE
tff(fact_475_linorder__neqE,axiom,
! [Xb: num,Y: num] :
( ( Xb != Y )
=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).
% linorder_neqE
tff(fact_476_linorder__neqE,axiom,
! [Xb: nat,Y: nat] :
( ( Xb != Y )
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).
% linorder_neqE
tff(fact_477_linorder__neqE,axiom,
! [Xb: int,Y: int] :
( ( Xb != Y )
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).
% linorder_neqE
tff(fact_478_order__less__asym,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).
% order_less_asym
tff(fact_479_order__less__asym,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).
% order_less_asym
tff(fact_480_order__less__asym,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).
% order_less_asym
tff(fact_481_order__less__asym,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).
% order_less_asym
tff(fact_482_order__less__asym,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).
% order_less_asym
tff(fact_483_linorder__neq__iff,axiom,
! [Xb: real,Y: real] :
( ( Xb != Y )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
| aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).
% linorder_neq_iff
tff(fact_484_linorder__neq__iff,axiom,
! [Xb: rat,Y: rat] :
( ( Xb != Y )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).
% linorder_neq_iff
tff(fact_485_linorder__neq__iff,axiom,
! [Xb: num,Y: num] :
( ( Xb != Y )
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
| aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).
% linorder_neq_iff
tff(fact_486_linorder__neq__iff,axiom,
! [Xb: nat,Y: nat] :
( ( Xb != Y )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).
% linorder_neq_iff
tff(fact_487_linorder__neq__iff,axiom,
! [Xb: int,Y: int] :
( ( Xb != Y )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).
% linorder_neq_iff
tff(fact_488_order__less__asym_H,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).
% order_less_asym'
tff(fact_489_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).
% order_less_asym'
tff(fact_490_order__less__asym_H,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).
% order_less_asym'
tff(fact_491_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).
% order_less_asym'
tff(fact_492_order__less__asym_H,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).
% order_less_asym'
tff(fact_493_order__less__trans,axiom,
! [Xb: real,Y: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).
% order_less_trans
tff(fact_494_order__less__trans,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).
% order_less_trans
tff(fact_495_order__less__trans,axiom,
! [Xb: num,Y: num,Z: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).
% order_less_trans
tff(fact_496_order__less__trans,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).
% order_less_trans
tff(fact_497_order__less__trans,axiom,
! [Xb: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).
% order_less_trans
tff(fact_498_ord__eq__less__subst,axiom,
! [A: real,F: fun_real_real,B: real,C: real] :
( ( A = aa_real_real(F,B) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_499_ord__eq__less__subst,axiom,
! [A: rat,F: fun_real_rat,B: real,C: real] :
( ( A = aa_real_rat(F,B) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_500_ord__eq__less__subst,axiom,
! [A: num,F: fun_real_num,B: real,C: real] :
( ( A = aa_real_num(F,B) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_501_ord__eq__less__subst,axiom,
! [A: nat,F: fun_real_nat,B: real,C: real] :
( ( A = aa_real_nat(F,B) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_502_ord__eq__less__subst,axiom,
! [A: int,F: fun_real_int,B: real,C: real] :
( ( A = aa_real_int(F,B) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_503_ord__eq__less__subst,axiom,
! [A: real,F: fun_rat_real,B: rat,C: rat] :
( ( A = aa_rat_real(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_504_ord__eq__less__subst,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( ( A = aa_rat_rat(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_505_ord__eq__less__subst,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( ( A = aa_rat_num(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_506_ord__eq__less__subst,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( ( A = aa_rat_nat(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_507_ord__eq__less__subst,axiom,
! [A: int,F: fun_rat_int,B: rat,C: rat] :
( ( A = aa_rat_int(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_508_ord__less__eq__subst,axiom,
! [A: real,B: real,F: fun_real_real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_real(F,B) = C )
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_509_ord__less__eq__subst,axiom,
! [A: real,B: real,F: fun_real_rat,C: rat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_rat(F,B) = C )
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_510_ord__less__eq__subst,axiom,
! [A: real,B: real,F: fun_real_num,C: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_num(F,B) = C )
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_511_ord__less__eq__subst,axiom,
! [A: real,B: real,F: fun_real_nat,C: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_nat(F,B) = C )
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_512_ord__less__eq__subst,axiom,
! [A: real,B: real,F: fun_real_int,C: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_int(F,B) = C )
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_513_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( ( aa_rat_real(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_514_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( ( aa_rat_rat(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_515_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( ( aa_rat_num(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_516_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( ( aa_rat_nat(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_517_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( ( aa_rat_int(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_518_order__less__irrefl,axiom,
! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).
% order_less_irrefl
tff(fact_519_order__less__irrefl,axiom,
! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).
% order_less_irrefl
tff(fact_520_order__less__irrefl,axiom,
! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).
% order_less_irrefl
tff(fact_521_order__less__irrefl,axiom,
! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).
% order_less_irrefl
tff(fact_522_order__less__irrefl,axiom,
! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).
% order_less_irrefl
tff(fact_523_order__less__subst1,axiom,
! [A: real,F: fun_real_real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_524_order__less__subst1,axiom,
! [A: real,F: fun_rat_real,B: rat,C: rat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_525_order__less__subst1,axiom,
! [A: real,F: fun_num_real,B: num,C: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_526_order__less__subst1,axiom,
! [A: real,F: fun_nat_real,B: nat,C: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_527_order__less__subst1,axiom,
! [A: real,F: fun_int_real,B: int,C: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_528_order__less__subst1,axiom,
! [A: rat,F: fun_real_rat,B: real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_529_order__less__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_530_order__less__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_531_order__less__subst1,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_532_order__less__subst1,axiom,
! [A: rat,F: fun_int_rat,B: int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_533_order__less__subst2,axiom,
! [A: real,B: real,F: fun_real_real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_534_order__less__subst2,axiom,
! [A: real,B: real,F: fun_real_rat,C: rat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_535_order__less__subst2,axiom,
! [A: real,B: real,F: fun_real_num,C: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_536_order__less__subst2,axiom,
! [A: real,B: real,F: fun_real_nat,C: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_537_order__less__subst2,axiom,
! [A: real,B: real,F: fun_real_int,C: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_538_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_539_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_540_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_541_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_542_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_543_order__less__not__sym,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).
% order_less_not_sym
tff(fact_544_order__less__not__sym,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).
% order_less_not_sym
tff(fact_545_order__less__not__sym,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).
% order_less_not_sym
tff(fact_546_order__less__not__sym,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).
% order_less_not_sym
tff(fact_547_order__less__not__sym,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).
% order_less_not_sym
tff(fact_548_order__less__imp__triv,axiom,
! [Xb: real,Y: real,P: $o] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_549_order__less__imp__triv,axiom,
! [Xb: rat,Y: rat,P: $o] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_550_order__less__imp__triv,axiom,
! [Xb: num,Y: num,P: $o] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_551_order__less__imp__triv,axiom,
! [Xb: nat,Y: nat,P: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_552_order__less__imp__triv,axiom,
! [Xb: int,Y: int,P: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_553_linorder__less__linear,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
| ( Xb = Y )
| aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).
% linorder_less_linear
tff(fact_554_linorder__less__linear,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
| ( Xb = Y )
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).
% linorder_less_linear
tff(fact_555_linorder__less__linear,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
| ( Xb = Y )
| aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).
% linorder_less_linear
tff(fact_556_linorder__less__linear,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
| ( Xb = Y )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).
% linorder_less_linear
tff(fact_557_linorder__less__linear,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
| ( Xb = Y )
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).
% linorder_less_linear
tff(fact_558_order__less__imp__not__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( Xb != Y ) ) ).
% order_less_imp_not_eq
tff(fact_559_order__less__imp__not__eq,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( Xb != Y ) ) ).
% order_less_imp_not_eq
tff(fact_560_order__less__imp__not__eq,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( Xb != Y ) ) ).
% order_less_imp_not_eq
tff(fact_561_order__less__imp__not__eq,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( Xb != Y ) ) ).
% order_less_imp_not_eq
tff(fact_562_order__less__imp__not__eq,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( Xb != Y ) ) ).
% order_less_imp_not_eq
tff(fact_563_order__less__imp__not__eq2,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( Y != Xb ) ) ).
% order_less_imp_not_eq2
tff(fact_564_order__less__imp__not__eq2,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( Y != Xb ) ) ).
% order_less_imp_not_eq2
tff(fact_565_order__less__imp__not__eq2,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( Y != Xb ) ) ).
% order_less_imp_not_eq2
tff(fact_566_order__less__imp__not__eq2,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( Y != Xb ) ) ).
% order_less_imp_not_eq2
tff(fact_567_order__less__imp__not__eq2,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( Y != Xb ) ) ).
% order_less_imp_not_eq2
tff(fact_568_order__less__imp__not__less,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).
% order_less_imp_not_less
tff(fact_569_order__less__imp__not__less,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).
% order_less_imp_not_less
tff(fact_570_order__less__imp__not__less,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).
% order_less_imp_not_less
tff(fact_571_order__less__imp__not__less,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).
% order_less_imp_not_less
tff(fact_572_order__less__imp__not__less,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).
% order_less_imp_not_less
tff(fact_573_linorder__neqE__nat,axiom,
! [Xb: nat,Y: nat] :
( ( Xb != Y )
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).
% linorder_neqE_nat
tff(fact_574_infinite__descent,axiom,
! [P: fun_nat_o,Nb: nat] :
( ! [N: nat] :
( ~ aa_nat_o(P,N)
=> ? [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
& ~ aa_nat_o(P,M3) ) )
=> aa_nat_o(P,Nb) ) ).
% infinite_descent
tff(fact_575_nat__less__induct,axiom,
! [P: fun_nat_o,Nb: nat] :
( ! [N: nat] :
( ! [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
=> aa_nat_o(P,M3) )
=> aa_nat_o(P,N) )
=> aa_nat_o(P,Nb) ) ).
% nat_less_induct
tff(fact_576_less__irrefl__nat,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).
% less_irrefl_nat
tff(fact_577_less__not__refl3,axiom,
! [S2: nat,Ta: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
=> ( S2 != Ta ) ) ).
% less_not_refl3
tff(fact_578_less__not__refl2,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
=> ( Mb != Nb ) ) ).
% less_not_refl2
tff(fact_579_less__not__refl,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).
% less_not_refl
tff(fact_580_nat__neq__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( Mb != Nb )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).
% nat_neq_iff
tff(fact_581_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_real,Nb: nat,Mb: nat] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_582_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_rat,Nb: nat,Mb: nat] :
( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_583_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_num,Nb: nat,Mb: nat] :
( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_584_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_nat,Nb: nat,Mb: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_585_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_int,Nb: nat,Mb: nat] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_586_lift__Suc__mono__less,axiom,
! [F: fun_nat_real,Nb: nat,N2: nat] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).
% lift_Suc_mono_less
tff(fact_587_lift__Suc__mono__less,axiom,
! [F: fun_nat_rat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).
% lift_Suc_mono_less
tff(fact_588_lift__Suc__mono__less,axiom,
! [F: fun_nat_num,Nb: nat,N2: nat] :
( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).
% lift_Suc_mono_less
tff(fact_589_lift__Suc__mono__less,axiom,
! [F: fun_nat_nat,Nb: nat,N2: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).
% lift_Suc_mono_less
tff(fact_590_lift__Suc__mono__less,axiom,
! [F: fun_nat_int,Nb: nat,N2: nat] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).
% lift_Suc_mono_less
tff(fact_591_order__le__imp__less__or__eq,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> ( aa_set_nat_o(ord_less_set_nat(Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_592_order__le__imp__less__or__eq,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_593_order__le__imp__less__or__eq,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_594_order__le__imp__less__or__eq,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_595_order__le__imp__less__or__eq,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_596_order__le__imp__less__or__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_597_linorder__le__less__linear,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).
% linorder_le_less_linear
tff(fact_598_linorder__le__less__linear,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
| aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).
% linorder_le_less_linear
tff(fact_599_linorder__le__less__linear,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).
% linorder_le_less_linear
tff(fact_600_linorder__le__less__linear,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).
% linorder_le_less_linear
tff(fact_601_linorder__le__less__linear,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
| aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).
% linorder_le_less_linear
tff(fact_602_order__less__le__subst2,axiom,
! [A: real,B: real,F: fun_real_rat,C: rat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_603_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_604_order__less__le__subst2,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_605_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_606_order__less__le__subst2,axiom,
! [A: int,B: int,F: fun_int_rat,C: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_607_order__less__le__subst2,axiom,
! [A: real,B: real,F: fun_real_num,C: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,B)),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_608_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_609_order__less__le__subst2,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_610_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_num,C: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_611_order__less__le__subst2,axiom,
! [A: int,B: int,F: fun_int_num,C: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,B)),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_int_num(F,X4)),aa_int_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_int_num(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_612_order__less__le__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_613_order__less__le__subst1,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_614_order__less__le__subst1,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_615_order__less__le__subst1,axiom,
! [A: int,F: fun_rat_int,B: rat,C: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_616_order__less__le__subst1,axiom,
! [A: real,F: fun_rat_real,B: rat,C: rat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_617_order__less__le__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_618_order__less__le__subst1,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_619_order__less__le__subst1,axiom,
! [A: nat,F: fun_num_nat,B: num,C: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_620_order__less__le__subst1,axiom,
! [A: int,F: fun_num_int,B: num,C: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_621_order__less__le__subst1,axiom,
! [A: real,F: fun_num_real,B: num,C: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_622_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_623_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_624_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_625_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_626_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_627_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_628_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_629_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_nat,C: nat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_630_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_int,C: int] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_631_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_real,C: real] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_632_order__le__less__subst1,axiom,
! [A: rat,F: fun_real_rat,B: real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_633_order__le__less__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_634_order__le__less__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_635_order__le__less__subst1,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_636_order__le__less__subst1,axiom,
! [A: rat,F: fun_int_rat,B: int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_637_order__le__less__subst1,axiom,
! [A: num,F: fun_real_num,B: real,C: real] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_real_num(F,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_638_order__le__less__subst1,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_639_order__le__less__subst1,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_640_order__le__less__subst1,axiom,
! [A: num,F: fun_nat_num,B: nat,C: nat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_nat_num(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_641_order__le__less__subst1,axiom,
! [A: num,F: fun_int_num,B: int,C: int] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,aa_int_num(F,X4)),aa_int_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_int_num(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_642_order__less__le__trans,axiom,
! [Xb: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(ord_less_set_nat(Xb),Y)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
=> aa_set_nat_o(ord_less_set_nat(Xb),Z) ) ) ).
% order_less_le_trans
tff(fact_643_order__less__le__trans,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).
% order_less_le_trans
tff(fact_644_order__less__le__trans,axiom,
! [Xb: num,Y: num,Z: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).
% order_less_le_trans
tff(fact_645_order__less__le__trans,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).
% order_less_le_trans
tff(fact_646_order__less__le__trans,axiom,
! [Xb: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).
% order_less_le_trans
tff(fact_647_order__less__le__trans,axiom,
! [Xb: real,Y: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).
% order_less_le_trans
tff(fact_648_order__le__less__trans,axiom,
! [Xb: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> ( aa_set_nat_o(ord_less_set_nat(Y),Z)
=> aa_set_nat_o(ord_less_set_nat(Xb),Z) ) ) ).
% order_le_less_trans
tff(fact_649_order__le__less__trans,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).
% order_le_less_trans
tff(fact_650_order__le__less__trans,axiom,
! [Xb: num,Y: num,Z: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).
% order_le_less_trans
tff(fact_651_order__le__less__trans,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).
% order_le_less_trans
tff(fact_652_order__le__less__trans,axiom,
! [Xb: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).
% order_le_less_trans
tff(fact_653_order__le__less__trans,axiom,
! [Xb: real,Y: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).
% order_le_less_trans
tff(fact_654_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).
% order_neq_le_trans
tff(fact_655_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% order_neq_le_trans
tff(fact_656_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).
% order_neq_le_trans
tff(fact_657_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% order_neq_le_trans
tff(fact_658_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% order_neq_le_trans
tff(fact_659_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% order_neq_le_trans
tff(fact_660_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> ( ( A != B )
=> aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).
% order_le_neq_trans
tff(fact_661_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( A != B )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% order_le_neq_trans
tff(fact_662_order__le__neq__trans,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( A != B )
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).
% order_le_neq_trans
tff(fact_663_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( A != B )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% order_le_neq_trans
tff(fact_664_order__le__neq__trans,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( ( A != B )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% order_le_neq_trans
tff(fact_665_order__le__neq__trans,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( ( A != B )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% order_le_neq_trans
tff(fact_666_order__less__imp__le,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_set_nat(Xb),Y)
=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).
% order_less_imp_le
tff(fact_667_order__less__imp__le,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).
% order_less_imp_le
tff(fact_668_order__less__imp__le,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).
% order_less_imp_le
tff(fact_669_order__less__imp__le,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).
% order_less_imp_le
tff(fact_670_order__less__imp__le,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).
% order_less_imp_le
tff(fact_671_order__less__imp__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).
% order_less_imp_le
tff(fact_672_linorder__not__less,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).
% linorder_not_less
tff(fact_673_linorder__not__less,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).
% linorder_not_less
tff(fact_674_linorder__not__less,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).
% linorder_not_less
tff(fact_675_linorder__not__less,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).
% linorder_not_less
tff(fact_676_linorder__not__less,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).
% linorder_not_less
tff(fact_677_linorder__not__le,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).
% linorder_not_le
tff(fact_678_linorder__not__le,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).
% linorder_not_le
tff(fact_679_linorder__not__le,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).
% linorder_not_le
tff(fact_680_linorder__not__le,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).
% linorder_not_le
tff(fact_681_linorder__not__le,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).
% linorder_not_le
tff(fact_682_order__less__le,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_set_nat(Xb),Y)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
& ( Xb != Y ) ) ) ).
% order_less_le
tff(fact_683_order__less__le,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
& ( Xb != Y ) ) ) ).
% order_less_le
tff(fact_684_order__less__le,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
& ( Xb != Y ) ) ) ).
% order_less_le
tff(fact_685_order__less__le,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
& ( Xb != Y ) ) ) ).
% order_less_le
tff(fact_686_order__less__le,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
& ( Xb != Y ) ) ) ).
% order_less_le
tff(fact_687_order__less__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
& ( Xb != Y ) ) ) ).
% order_less_le
tff(fact_688_order__le__less,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
<=> ( aa_set_nat_o(ord_less_set_nat(Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_less
tff(fact_689_order__le__less,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_less
tff(fact_690_order__le__less,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_less
tff(fact_691_order__le__less,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_less
tff(fact_692_order__le__less,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_less
tff(fact_693_order__le__less,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
| ( Xb = Y ) ) ) ).
% order_le_less
tff(fact_694_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(ord_less_set_nat(B),A)
=> aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).
% dual_order.strict_implies_order
tff(fact_695_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_696_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_697_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_698_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_699_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_700_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A),B)
=> aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).
% order.strict_implies_order
tff(fact_701_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% order.strict_implies_order
tff(fact_702_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).
% order.strict_implies_order
tff(fact_703_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% order.strict_implies_order
tff(fact_704_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% order.strict_implies_order
tff(fact_705_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% order.strict_implies_order
tff(fact_706_dual__order_Ostrict__iff__not,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(ord_less_set_nat(B),A)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
& ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_707_dual__order_Ostrict__iff__not,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
& ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_708_dual__order_Ostrict__iff__not,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
& ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_709_dual__order_Ostrict__iff__not,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_710_dual__order_Ostrict__iff__not,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_711_dual__order_Ostrict__iff__not,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
& ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_712_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_set_nat(B),A)
=> ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
=> aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_713_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_714_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_715_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_716_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_717_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_718_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(B),A)
=> ( aa_set_nat_o(ord_less_set_nat(C),B)
=> aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_719_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_720_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_721_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_722_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_723_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_724_dual__order_Ostrict__iff__order,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(ord_less_set_nat(B),A)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_725_dual__order_Ostrict__iff__order,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_726_dual__order_Ostrict__iff__order,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_727_dual__order_Ostrict__iff__order,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_728_dual__order_Ostrict__iff__order,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_729_dual__order_Ostrict__iff__order,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_730_dual__order_Oorder__iff__strict,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(B),A)
<=> ( aa_set_nat_o(ord_less_set_nat(B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_731_dual__order_Oorder__iff__strict,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_732_dual__order_Oorder__iff__strict,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_733_dual__order_Oorder__iff__strict,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_734_dual__order_Oorder__iff__strict,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_735_dual__order_Oorder__iff__strict,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_736_dense__le__bounded,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( ! [W2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),W2)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).
% dense_le_bounded
tff(fact_737_dense__le__bounded,axiom,
! [Xb: real,Y: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( ! [W2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),W2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).
% dense_le_bounded
tff(fact_738_dense__ge__bounded,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
=> ( ! [W2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).
% dense_ge_bounded
tff(fact_739_dense__ge__bounded,axiom,
! [Z: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
=> ( ! [W2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W2) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).
% dense_ge_bounded
tff(fact_740_order_Ostrict__iff__not,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A),B)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
& ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).
% order.strict_iff_not
tff(fact_741_order_Ostrict__iff__not,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
& ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).
% order.strict_iff_not
tff(fact_742_order_Ostrict__iff__not,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
& ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).
% order.strict_iff_not
tff(fact_743_order_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).
% order.strict_iff_not
tff(fact_744_order_Ostrict__iff__not,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% order.strict_iff_not
tff(fact_745_order_Ostrict__iff__not,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
& ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).
% order.strict_iff_not
tff(fact_746_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A),B)
=> ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
=> aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).
% order.strict_trans2
tff(fact_747_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).
% order.strict_trans2
tff(fact_748_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).
% order.strict_trans2
tff(fact_749_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% order.strict_trans2
tff(fact_750_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% order.strict_trans2
tff(fact_751_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).
% order.strict_trans2
tff(fact_752_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> ( aa_set_nat_o(ord_less_set_nat(B),C)
=> aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).
% order.strict_trans1
tff(fact_753_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).
% order.strict_trans1
tff(fact_754_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).
% order.strict_trans1
tff(fact_755_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% order.strict_trans1
tff(fact_756_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% order.strict_trans1
tff(fact_757_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).
% order.strict_trans1
tff(fact_758_order_Ostrict__iff__order,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A),B)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_759_order_Ostrict__iff__order,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_760_order_Ostrict__iff__order,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_761_order_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_762_order_Ostrict__iff__order,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_763_order_Ostrict__iff__order,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_764_order_Oorder__iff__strict,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),B)
<=> ( aa_set_nat_o(ord_less_set_nat(A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_765_order_Oorder__iff__strict,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_766_order_Oorder__iff__strict,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_767_order_Oorder__iff__strict,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_768_order_Oorder__iff__strict,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_769_order_Oorder__iff__strict,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_770_not__le__imp__less,axiom,
! [Y: rat,Xb: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).
% not_le_imp_less
tff(fact_771_not__le__imp__less,axiom,
! [Y: num,Xb: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).
% not_le_imp_less
tff(fact_772_not__le__imp__less,axiom,
! [Y: nat,Xb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).
% not_le_imp_less
tff(fact_773_not__le__imp__less,axiom,
! [Y: int,Xb: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).
% not_le_imp_less
tff(fact_774_not__le__imp__less,axiom,
! [Y: real,Xb: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% not_le_imp_less
tff(fact_775_less__le__not__le,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_set_nat(Xb),Y)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
& ~ aa_set_nat_o(ord_less_eq_set_nat(Y),Xb) ) ) ).
% less_le_not_le
tff(fact_776_less__le__not__le,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
& ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).
% less_le_not_le
tff(fact_777_less__le__not__le,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
& ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).
% less_le_not_le
tff(fact_778_less__le__not__le,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).
% less_le_not_le
tff(fact_779_less__le__not__le,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).
% less_le_not_le
tff(fact_780_less__le__not__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
& ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).
% less_le_not_le
tff(fact_781_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Z) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).
% dense_le
tff(fact_782_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Z) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).
% dense_le
tff(fact_783_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X4) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).
% dense_ge
tff(fact_784_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X4) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).
% dense_ge
tff(fact_785_antisym__conv2,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> ( ~ aa_set_nat_o(ord_less_set_nat(Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv2
tff(fact_786_antisym__conv2,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv2
tff(fact_787_antisym__conv2,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv2
tff(fact_788_antisym__conv2,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv2
tff(fact_789_antisym__conv2,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv2
tff(fact_790_antisym__conv2,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv2
tff(fact_791_antisym__conv1,axiom,
! [Xb: set_nat,Y: set_nat] :
( ~ aa_set_nat_o(ord_less_set_nat(Xb),Y)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv1
tff(fact_792_antisym__conv1,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv1
tff(fact_793_antisym__conv1,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv1
tff(fact_794_antisym__conv1,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv1
tff(fact_795_antisym__conv1,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv1
tff(fact_796_antisym__conv1,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% antisym_conv1
tff(fact_797_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ~ aa_set_nat_o(ord_less_set_nat(A),B)
<=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_798_nless__le,axiom,
! [A: rat,B: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
<=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_799_nless__le,axiom,
! [A: num,B: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
<=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_800_nless__le,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_801_nless__le,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_802_nless__le,axiom,
! [A: real,B: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
<=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_803_leI,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).
% leI
tff(fact_804_leI,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).
% leI
tff(fact_805_leI,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).
% leI
tff(fact_806_leI,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).
% leI
tff(fact_807_leI,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).
% leI
tff(fact_808_leD,axiom,
! [Y: set_nat,Xb: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Y),Xb)
=> ~ aa_set_nat_o(ord_less_set_nat(Xb),Y) ) ).
% leD
tff(fact_809_leD,axiom,
! [Y: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).
% leD
tff(fact_810_leD,axiom,
! [Y: num,Xb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).
% leD
tff(fact_811_leD,axiom,
! [Y: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).
% leD
tff(fact_812_leD,axiom,
! [Y: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).
% leD
tff(fact_813_leD,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% leD
tff(fact_814_verit__comp__simplify1_I3_J,axiom,
! [B2: rat,A2: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).
% verit_comp_simplify1(3)
tff(fact_815_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A2: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).
% verit_comp_simplify1(3)
tff(fact_816_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).
% verit_comp_simplify1(3)
tff(fact_817_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).
% verit_comp_simplify1(3)
tff(fact_818_verit__comp__simplify1_I3_J,axiom,
! [B2: real,A2: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).
% verit_comp_simplify1(3)
tff(fact_819_size__neq__size__imp__neq,axiom,
! [Xb: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Y) )
=> ( Xb != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_820_size__neq__size__imp__neq,axiom,
! [Xb: list_o,Y: list_o] :
( ( size_size_list_o(Xb) != size_size_list_o(Y) )
=> ( Xb != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_821_size__neq__size__imp__neq,axiom,
! [Xb: list_nat,Y: list_nat] :
( ( size_size_list_nat(Xb) != size_size_list_nat(Y) )
=> ( Xb != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_822_size__neq__size__imp__neq,axiom,
! [Xb: list_int,Y: list_int] :
( ( size_size_list_int(Xb) != size_size_list_int(Y) )
=> ( Xb != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_823_size__neq__size__imp__neq,axiom,
! [Xb: num,Y: num] :
( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
=> ( Xb != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_824_less__numeral__extra_I4_J,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).
% less_numeral_extra(4)
tff(fact_825_less__numeral__extra_I4_J,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).
% less_numeral_extra(4)
tff(fact_826_less__numeral__extra_I4_J,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).
% less_numeral_extra(4)
tff(fact_827_less__numeral__extra_I4_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).
% less_numeral_extra(4)
tff(fact_828_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% add_less_imp_less_right
tff(fact_829_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% add_less_imp_less_right
tff(fact_830_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_imp_less_right
tff(fact_831_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_imp_less_right
tff(fact_832_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% add_less_imp_less_left
tff(fact_833_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% add_less_imp_less_left
tff(fact_834_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_imp_less_left
tff(fact_835_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_imp_less_left
tff(fact_836_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_837_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).
% add_strict_right_mono
tff(fact_838_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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_839_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,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_840_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_841_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).
% add_strict_left_mono
tff(fact_842_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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_843_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,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_844_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
=> aa_real_o(aa_real_fun_real_o(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_845_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).
% add_strict_mono
tff(fact_846_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
=> aa_nat_o(aa_nat_fun_nat_o(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_847_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
=> aa_int_o(aa_int_fun_int_o(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_848_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
& ( K = L ) )
=> aa_real_o(aa_real_fun_real_o(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_849_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
& ( K = L ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_850_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& ( K = L ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_851_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
& ( K = L ) )
=> aa_int_o(aa_int_fun_int_o(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_852_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
=> aa_real_o(aa_real_fun_real_o(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_853_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_854_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_855_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(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_856_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
& aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
=> aa_real_o(aa_real_fun_real_o(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_857_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_858_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_859_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(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_860_not__less__less__Suc__eq,axiom,
! [Nb: nat,Mb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
<=> ( Nb = Mb ) ) ) ).
% not_less_less_Suc_eq
tff(fact_861_strict__inc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( ! [I3: nat] :
( ( J = aa_nat_nat(suc,I3) )
=> aa_nat_o(P,I3) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
=> ( aa_nat_o(P,aa_nat_nat(suc,I3))
=> aa_nat_o(P,I3) ) )
=> aa_nat_o(P,I) ) ) ) ).
% strict_inc_induct
tff(fact_862_less__Suc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
=> ( ! [I3: nat,J2: nat,K3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K3)
=> ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
=> ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K3)
=> aa_nat_o(aa_nat_fun_nat_o(P,I3),K3) ) ) ) )
=> aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).
% less_Suc_induct
tff(fact_863_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).
% less_trans_Suc
tff(fact_864_Suc__less__SucD,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% Suc_less_SucD
tff(fact_865_less__antisym,axiom,
! [Nb: nat,Mb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
=> ( Mb = Nb ) ) ) ).
% less_antisym
tff(fact_866_Suc__less__eq2,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
<=> ? [M5: nat] :
( ( Mb = aa_nat_nat(suc,M5) )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M5) ) ) ).
% Suc_less_eq2
tff(fact_867_All__less__Suc,axiom,
! [Nb: nat,P: fun_nat_o] :
( ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
=> aa_nat_o(P,I4) )
<=> ( aa_nat_o(P,Nb)
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
=> aa_nat_o(P,I4) ) ) ) ).
% All_less_Suc
tff(fact_868_not__less__eq,axiom,
! [Mb: nat,Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).
% not_less_eq
tff(fact_869_less__Suc__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| ( Mb = Nb ) ) ) ).
% less_Suc_eq
tff(fact_870_Ex__less__Suc,axiom,
! [Nb: nat,P: fun_nat_o] :
( ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
& aa_nat_o(P,I4) )
<=> ( aa_nat_o(P,Nb)
| ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
& aa_nat_o(P,I4) ) ) ) ).
% Ex_less_Suc
tff(fact_871_less__SucI,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).
% less_SucI
tff(fact_872_less__SucE,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( Mb = Nb ) ) ) ).
% less_SucE
tff(fact_873_Suc__lessI,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( ( aa_nat_nat(suc,Mb) != Nb )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).
% Suc_lessI
tff(fact_874_Suc__lessE,axiom,
! [I: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
=> ~ ! [J2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
=> ( K != aa_nat_nat(suc,J2) ) ) ) ).
% Suc_lessE
tff(fact_875_Suc__lessD,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% Suc_lessD
tff(fact_876_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
=> ( ( K != aa_nat_nat(suc,I) )
=> ~ ! [J2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
=> ( K != aa_nat_nat(suc,J2) ) ) ) ) ).
% Nat.lessE
tff(fact_877_less__mono__imp__le__mono,axiom,
! [F: fun_nat_nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).
% less_mono_imp_le_mono
tff(fact_878_le__neq__implies__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( ( Mb != Nb )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% le_neq_implies_less
tff(fact_879_less__or__eq__imp__le,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| ( Mb = Nb ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% less_or_eq_imp_le
tff(fact_880_le__eq__less__or__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| ( Mb = Nb ) ) ) ).
% le_eq_less_or_eq
tff(fact_881_less__imp__le__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% less_imp_le_nat
tff(fact_882_nat__less__le,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& ( Mb != Nb ) ) ) ).
% nat_less_le
tff(fact_883_less__add__eq__less,axiom,
! [K: nat,L: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% less_add_eq_less
tff(fact_884_trans__less__add2,axiom,
! [I: nat,J: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).
% trans_less_add2
tff(fact_885_trans__less__add1,axiom,
! [I: nat,J: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).
% trans_less_add1
tff(fact_886_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).
% add_less_mono1
tff(fact_887_not__add__less2,axiom,
! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).
% not_add_less2
tff(fact_888_not__add__less1,axiom,
! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).
% not_add_less1
tff(fact_889_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).
% add_less_mono
tff(fact_890_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).
% add_lessD1
tff(fact_891_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).
% add_less_le_mono
tff(fact_892_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
=> aa_nat_o(aa_nat_fun_nat_o(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_893_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
=> aa_int_o(aa_int_fun_int_o(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_894_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
=> aa_real_o(aa_real_fun_real_o(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_895_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).
% add_le_less_mono
tff(fact_896_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
=> aa_nat_o(aa_nat_fun_nat_o(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_897_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
=> aa_int_o(aa_int_fun_int_o(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_898_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
=> aa_real_o(aa_real_fun_real_o(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_899_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_900_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_901_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_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_902_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
=> aa_real_o(aa_real_fun_real_o(ord_less_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_903_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_904_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_905_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(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_906_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
& aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
=> aa_real_o(aa_real_fun_real_o(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_907_not__numeral__less__one,axiom,
! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).
% not_numeral_less_one
tff(fact_908_not__numeral__less__one,axiom,
! [Nb: num] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Nb)),one_on7984719198319812577d_enat) ).
% not_numeral_less_one
tff(fact_909_not__numeral__less__one,axiom,
! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).
% not_numeral_less_one
tff(fact_910_not__numeral__less__one,axiom,
! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).
% not_numeral_less_one
tff(fact_911_not__numeral__less__one,axiom,
! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).
% not_numeral_less_one
tff(fact_912_Suc__leI,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).
% Suc_leI
tff(fact_913_Suc__le__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% Suc_le_eq
tff(fact_914_dec__induct,axiom,
! [I: nat,J: nat,P: fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(P,I)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
=> ( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
=> aa_nat_o(P,J) ) ) ) ).
% dec_induct
tff(fact_915_inc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(P,J)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
=> ( aa_nat_o(P,aa_nat_nat(suc,N))
=> aa_nat_o(P,N) ) ) )
=> aa_nat_o(P,I) ) ) ) ).
% inc_induct
tff(fact_916_Suc__le__lessD,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% Suc_le_lessD
tff(fact_917_le__less__Suc__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
<=> ( Nb = Mb ) ) ) ).
% le_less_Suc_eq
tff(fact_918_less__Suc__eq__le,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% less_Suc_eq_le
tff(fact_919_less__eq__Suc__le,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).
% less_eq_Suc_le
tff(fact_920_le__imp__less__Suc,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).
% le_imp_less_Suc
tff(fact_921_less__natE,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ~ ! [Q2: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ) ).
% less_natE
tff(fact_922_less__add__Suc1,axiom,
! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb))) ).
% less_add_Suc1
tff(fact_923_less__add__Suc2,axiom,
! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),I))) ).
% less_add_Suc2
tff(fact_924_less__iff__Suc__add,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
<=> ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).
% less_iff_Suc_add
tff(fact_925_less__imp__Suc__add,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).
% less_imp_Suc_add
tff(fact_926_dbl__def,axiom,
! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb) ).
% dbl_def
tff(fact_927_dbl__def,axiom,
! [Xb: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xb) = aa_rat_rat(plus_plus_rat(Xb),Xb) ).
% dbl_def
tff(fact_928_dbl__def,axiom,
! [Xb: int] : aa_int_int(neg_numeral_dbl_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb) ).
% dbl_def
tff(fact_929_mono__nat__linear__lb,axiom,
! [F: fun_nat_nat,Mb: nat,K: nat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))) ) ).
% mono_nat_linear_lb
tff(fact_930_order__antisym__conv,axiom,
! [Y: set_nat,Xb: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Y),Xb)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
<=> ( Xb = Y ) ) ) ).
% order_antisym_conv
tff(fact_931_order__antisym__conv,axiom,
! [Y: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% order_antisym_conv
tff(fact_932_order__antisym__conv,axiom,
! [Y: num,Xb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% order_antisym_conv
tff(fact_933_order__antisym__conv,axiom,
! [Y: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% order_antisym_conv
tff(fact_934_order__antisym__conv,axiom,
! [Y: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% order_antisym_conv
tff(fact_935_order__antisym__conv,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
<=> ( Xb = Y ) ) ) ).
% order_antisym_conv
tff(fact_936_linorder__le__cases,axiom,
! [Xb: rat,Y: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).
% linorder_le_cases
tff(fact_937_linorder__le__cases,axiom,
! [Xb: num,Y: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).
% linorder_le_cases
tff(fact_938_linorder__le__cases,axiom,
! [Xb: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).
% linorder_le_cases
tff(fact_939_linorder__le__cases,axiom,
! [Xb: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).
% linorder_le_cases
tff(fact_940_linorder__le__cases,axiom,
! [Xb: real,Y: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).
% linorder_le_cases
tff(fact_941_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( aa_rat_rat(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_942_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( aa_rat_num(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_943_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( aa_rat_nat(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_944_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( aa_rat_int(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_945_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( aa_rat_real(F,B) = C )
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_946_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( aa_num_rat(F,B) = C )
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_947_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( aa_num_num(F,B) = C )
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_948_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_nat,C: nat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( aa_num_nat(F,B) = C )
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_949_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_int,C: int] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( aa_num_int(F,B) = C )
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_950_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_real,C: real] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( aa_num_real(F,B) = C )
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_951_ord__eq__le__subst,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( ( A = aa_rat_rat(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_952_ord__eq__le__subst,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( ( A = aa_rat_num(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_953_ord__eq__le__subst,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( ( A = aa_rat_nat(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_954_ord__eq__le__subst,axiom,
! [A: int,F: fun_rat_int,B: rat,C: rat] :
( ( A = aa_rat_int(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_955_ord__eq__le__subst,axiom,
! [A: real,F: fun_rat_real,B: rat,C: rat] :
( ( A = aa_rat_real(F,B) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_956_ord__eq__le__subst,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( ( A = aa_num_rat(F,B) )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_957_ord__eq__le__subst,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( ( A = aa_num_num(F,B) )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_958_ord__eq__le__subst,axiom,
! [A: nat,F: fun_num_nat,B: num,C: num] :
( ( A = aa_num_nat(F,B) )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_959_ord__eq__le__subst,axiom,
! [A: int,F: fun_num_int,B: num,C: num] :
( ( A = aa_num_int(F,B) )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_960_ord__eq__le__subst,axiom,
! [A: real,F: fun_num_real,B: num,C: num] :
( ( A = aa_num_real(F,B) )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_961_linorder__linear,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).
% linorder_linear
tff(fact_962_linorder__linear,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
| aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).
% linorder_linear
tff(fact_963_linorder__linear,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).
% linorder_linear
tff(fact_964_linorder__linear,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).
% linorder_linear
tff(fact_965_linorder__linear,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
| aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).
% linorder_linear
tff(fact_966_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
| ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).
% verit_la_disequality
tff(fact_967_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
| ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).
% verit_la_disequality
tff(fact_968_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
| ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).
% verit_la_disequality
tff(fact_969_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
| ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% verit_la_disequality
tff(fact_970_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
| ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).
% verit_la_disequality
tff(fact_971_order__eq__refl,axiom,
! [Xb: set_nat,Y: set_nat] :
( ( Xb = Y )
=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).
% order_eq_refl
tff(fact_972_order__eq__refl,axiom,
! [Xb: rat,Y: rat] :
( ( Xb = Y )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).
% order_eq_refl
tff(fact_973_order__eq__refl,axiom,
! [Xb: num,Y: num] :
( ( Xb = Y )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).
% order_eq_refl
tff(fact_974_order__eq__refl,axiom,
! [Xb: nat,Y: nat] :
( ( Xb = Y )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).
% order_eq_refl
tff(fact_975_order__eq__refl,axiom,
! [Xb: int,Y: int] :
( ( Xb = Y )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).
% order_eq_refl
tff(fact_976_order__eq__refl,axiom,
! [Xb: real,Y: real] :
( ( Xb = Y )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).
% order_eq_refl
tff(fact_977_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_978_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_979_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_980_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_981_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_982_order__subst2,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_983_order__subst2,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_984_order__subst2,axiom,
! [A: num,B: num,F: fun_num_nat,C: nat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_985_order__subst2,axiom,
! [A: num,B: num,F: fun_num_int,C: int] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_986_order__subst2,axiom,
! [A: num,B: num,F: fun_num_real,C: real] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_987_order__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).
% order_subst1
tff(fact_988_order__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).
% order_subst1
tff(fact_989_order__subst1,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).
% order_subst1
tff(fact_990_order__subst1,axiom,
! [A: rat,F: fun_int_rat,B: int,C: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).
% order_subst1
tff(fact_991_order__subst1,axiom,
! [A: rat,F: fun_real_rat,B: real,C: real] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,C)) ) ) ) ).
% order_subst1
tff(fact_992_order__subst1,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> ( ! [X4: rat,Y3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).
% order_subst1
tff(fact_993_order__subst1,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> ( ! [X4: num,Y3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).
% order_subst1
tff(fact_994_order__subst1,axiom,
! [A: num,F: fun_nat_num,B: nat,C: nat] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).
% order_subst1
tff(fact_995_order__subst1,axiom,
! [A: num,F: fun_int_num,B: int,C: int] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> ( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).
% order_subst1
tff(fact_996_order__subst1,axiom,
! [A: num,F: fun_real_num,B: real,C: real] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_real_num(F,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> ( ! [X4: real,Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,X4)),aa_real_num(F,Y3)) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_real_num(F,C)) ) ) ) ).
% order_subst1
tff(fact_997_Orderings_Oorder__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
<=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
& aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_998_Orderings_Oorder__eq__iff,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_999_Orderings_Oorder__eq__iff,axiom,
! [A: num,B: num] :
( ( A = B )
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1000_Orderings_Oorder__eq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1001_Orderings_Oorder__eq__iff,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1002_Orderings_Oorder__eq__iff,axiom,
! [A: real,B: real] :
( ( A = B )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1003_antisym,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1004_antisym,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1005_antisym,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1006_antisym,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1007_antisym,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1008_antisym,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1009_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(B),A)
=> ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
=> aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).
% dual_order.trans
tff(fact_1010_dual__order_Otrans,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).
% dual_order.trans
tff(fact_1011_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).
% dual_order.trans
tff(fact_1012_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).
% dual_order.trans
tff(fact_1013_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).
% dual_order.trans
tff(fact_1014_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A) ) ) ).
% dual_order.trans
tff(fact_1015_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(B),A)
=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1016_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1017_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1018_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1019_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1020_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1021_dual__order_Oeq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
<=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
& aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1022_dual__order_Oeq__iff,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1023_dual__order_Oeq__iff,axiom,
! [A: num,B: num] :
( ( A = B )
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1024_dual__order_Oeq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1025_dual__order_Oeq__iff,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1026_dual__order_Oeq__iff,axiom,
! [A: real,B: real] :
( ( A = B )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1027_linorder__wlog,axiom,
! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
( ! [A4: rat,B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A4),B4)
=> aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
=> ( ! [A4: rat,B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(P,B4),A4)
=> aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
=> aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1028_linorder__wlog,axiom,
! [P: fun_num_fun_num_o,A: num,B: num] :
( ! [A4: num,B4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A4),B4)
=> aa_num_o(aa_num_fun_num_o(P,A4),B4) )
=> ( ! [A4: num,B4: num] :
( aa_num_o(aa_num_fun_num_o(P,B4),A4)
=> aa_num_o(aa_num_fun_num_o(P,A4),B4) )
=> aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1029_linorder__wlog,axiom,
! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A4),B4)
=> aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
=> ( ! [A4: nat,B4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,B4),A4)
=> aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
=> aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1030_linorder__wlog,axiom,
! [P: fun_int_fun_int_o,A: int,B: int] :
( ! [A4: int,B4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B4)
=> aa_int_o(aa_int_fun_int_o(P,A4),B4) )
=> ( ! [A4: int,B4: int] :
( aa_int_o(aa_int_fun_int_o(P,B4),A4)
=> aa_int_o(aa_int_fun_int_o(P,A4),B4) )
=> aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1031_linorder__wlog,axiom,
! [P: fun_real_fun_real_o,A: real,B: real] :
( ! [A4: real,B4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),B4)
=> aa_real_o(aa_real_fun_real_o(P,A4),B4) )
=> ( ! [A4: real,B4: real] :
( aa_real_o(aa_real_fun_real_o(P,B4),A4)
=> aa_real_o(aa_real_fun_real_o(P,A4),B4) )
=> aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1032_order__trans,axiom,
! [Xb: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Z) ) ) ).
% order_trans
tff(fact_1033_order__trans,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) ) ) ).
% order_trans
tff(fact_1034_order__trans,axiom,
! [Xb: num,Y: num,Z: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) ) ) ).
% order_trans
tff(fact_1035_order__trans,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) ) ) ).
% order_trans
tff(fact_1036_order__trans,axiom,
! [Xb: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) ) ) ).
% order_trans
tff(fact_1037_order__trans,axiom,
! [Xb: real,Y: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Z) ) ) ).
% order_trans
tff(fact_1038_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
=> aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).
% order.trans
tff(fact_1039_order_Otrans,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).
% order.trans
tff(fact_1040_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).
% order.trans
tff(fact_1041_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% order.trans
tff(fact_1042_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% order.trans
tff(fact_1043_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C) ) ) ).
% order.trans
tff(fact_1044_order__antisym,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xb)
=> ( Xb = Y ) ) ) ).
% order_antisym
tff(fact_1045_order__antisym,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> ( Xb = Y ) ) ) ).
% order_antisym
tff(fact_1046_order__antisym,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
=> ( Xb = Y ) ) ) ).
% order_antisym
tff(fact_1047_order__antisym,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> ( Xb = Y ) ) ) ).
% order_antisym
tff(fact_1048_order__antisym,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
=> ( Xb = Y ) ) ) ).
% order_antisym
tff(fact_1049_order__antisym,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> ( Xb = Y ) ) ) ).
% order_antisym
tff(fact_1050_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),B)
=> ( ( B = C )
=> aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1051_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( ( B = C )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1052_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( ( B = C )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1053_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( B = C )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1054_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( ( B = C )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1055_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( ( B = C )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1056_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
=> aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1057_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1058_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1059_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1060_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1061_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1062_order__class_Oorder__eq__iff,axiom,
! [Xb: set_nat,Y: set_nat] :
( ( Xb = Y )
<=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
& aa_set_nat_o(ord_less_eq_set_nat(Y),Xb) ) ) ).
% order_class.order_eq_iff
tff(fact_1063_order__class_Oorder__eq__iff,axiom,
! [Xb: rat,Y: rat] :
( ( Xb = Y )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).
% order_class.order_eq_iff
tff(fact_1064_order__class_Oorder__eq__iff,axiom,
! [Xb: num,Y: num] :
( ( Xb = Y )
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).
% order_class.order_eq_iff
tff(fact_1065_order__class_Oorder__eq__iff,axiom,
! [Xb: nat,Y: nat] :
( ( Xb = Y )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).
% order_class.order_eq_iff
tff(fact_1066_order__class_Oorder__eq__iff,axiom,
! [Xb: int,Y: int] :
( ( Xb = Y )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).
% order_class.order_eq_iff
tff(fact_1067_order__class_Oorder__eq__iff,axiom,
! [Xb: real,Y: real] :
( ( Xb = Y )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).
% order_class.order_eq_iff
tff(fact_1068_le__cases3,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) )
=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) )
=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb) )
=> ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1069_le__cases3,axiom,
! [Xb: num,Y: num,Z: num] :
( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
=> ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) )
=> ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) )
=> ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) )
=> ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb) )
=> ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1070_le__cases3,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb) )
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1071_le__cases3,axiom,
! [Xb: int,Y: int,Z: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb) )
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1072_le__cases3,axiom,
! [Xb: real,Y: real,Z: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Z) )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Z)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Y) )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Y)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Xb) )
=> ~ ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Xb)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1073_nle__le,axiom,
! [A: rat,B: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1074_nle__le,axiom,
! [A: num,B: num] :
( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1075_nle__le,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1076_nle__le,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1077_nle__le,axiom,
! [A: real,B: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1078_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).
% verit_comp_simplify1(2)
tff(fact_1079_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).
% verit_comp_simplify1(2)
tff(fact_1080_verit__comp__simplify1_I2_J,axiom,
! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).
% verit_comp_simplify1(2)
tff(fact_1081_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).
% verit_comp_simplify1(2)
tff(fact_1082_verit__comp__simplify1_I2_J,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).
% verit_comp_simplify1(2)
tff(fact_1083_verit__comp__simplify1_I2_J,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),A) ).
% verit_comp_simplify1(2)
tff(fact_1084_verit__eq__simplify_I10_J,axiom,
! [X2: num] : one != aa_num_num(bit0,X2) ).
% verit_eq_simplify(10)
tff(fact_1085_maxt__corr,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
=> aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).
% maxt_corr
tff(fact_1086_maxt__sound,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
=> ( vEBT_vebt_maxt(Ta) = some_nat(Xb) ) ) ) ).
% maxt_sound
tff(fact_1087_power__increasing__iff,axiom,
! [B: rat,Xb: nat,Y: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).
% power_increasing_iff
tff(fact_1088_power__increasing__iff,axiom,
! [B: nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).
% power_increasing_iff
tff(fact_1089_power__increasing__iff,axiom,
! [B: int,Xb: nat,Y: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xb)),aa_nat_int(power_power_int(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).
% power_increasing_iff
tff(fact_1090_power__increasing__iff,axiom,
! [B: real,Xb: nat,Y: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).
% power_increasing_iff
tff(fact_1091_succ__min,axiom,
! [Dega: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Mia) ) ) ) ).
% succ_min
tff(fact_1092_pred__max,axiom,
! [Dega: nat,Maa: nat,Xb: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb)
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Maa) ) ) ) ).
% pred_max
tff(fact_1093_power__strict__increasing__iff,axiom,
! [B: real,Xb: nat,Y: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_1094_power__strict__increasing__iff,axiom,
! [B: rat,Xb: nat,Y: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_1095_power__strict__increasing__iff,axiom,
! [B: nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_1096_power__strict__increasing__iff,axiom,
! [B: int,Xb: nat,Y: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Xb)),aa_nat_int(power_power_int(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_1097_helpyd,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Y) )
=> aa_nat_o(aa_nat_fun_nat_o(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))),Nb)) ) ) ).
% helpyd
tff(fact_1098_helpypredd,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Y) )
=> aa_nat_o(aa_nat_fun_nat_o(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))),Nb)) ) ) ).
% helpypredd
tff(fact_1099_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
=> ? [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
& aa_nat_o(aa_nat_fun_nat_o(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))) ) ) ) ).
% ex_power_ivl2
tff(fact_1100_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
=> ? [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
& aa_nat_o(aa_nat_fun_nat_o(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))) ) ) ) ).
% ex_power_ivl1
tff(fact_1101_misiz,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( some_nat(Mb) = vEBT_vebt_mint(Ta) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).
% misiz
tff(fact_1102_inthall,axiom,
! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Nb: nat] :
( ! [X4: product_prod_nat_nat] :
( member8440522571783428010at_nat(X4,set_Pr5648618587558075414at_nat(Xs))
=> aa_Pro6150637604445964947_nat_o(P,X4) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s5460976970255530739at_nat(Xs))
=> aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,Nb)) ) ) ).
% inthall
tff(fact_1103_inthall,axiom,
! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),set_complex2(Xs))
=> aa_complex_o(P,X4) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
=> aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).
% inthall
tff(fact_1104_inthall,axiom,
! [Xs: list_real,P: fun_real_o,Nb: nat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),set_real2(Xs))
=> aa_real_o(P,X4) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
=> aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).
% inthall
tff(fact_1105_inthall,axiom,
! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs))
=> aa_VEBT_VEBT_o(P,X4) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
=> aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).
% inthall
tff(fact_1106_inthall,axiom,
! [Xs: list_o,P: fun_o_o,Nb: nat] :
( ! [X4: $o] :
( member_o((X4),set_o2(Xs))
=> aa_o_o(P,(X4)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
=> aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).
% inthall
tff(fact_1107_inthall,axiom,
! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
=> aa_nat_o(P,X4) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
=> aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).
% inthall
tff(fact_1108_inthall,axiom,
! [Xs: list_int,P: fun_int_o,Nb: nat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),set_int2(Xs))
=> aa_int_o(P,X4) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
=> aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).
% inthall
tff(fact_1109_set__vebt__set__vebt_H__valid,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).
% set_vebt_set_vebt'_valid
tff(fact_1110_mint__member,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
=> aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).
% mint_member
tff(fact_1111_local_Opower__def,axiom,
vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).
% local.power_def
tff(fact_1112_mint__corr__help,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
=> ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xb) ) ) ) ).
% mint_corr_help
tff(fact_1113_mint__sound,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
=> ( vEBT_vebt_mint(Ta) = some_nat(Xb) ) ) ) ).
% mint_sound
tff(fact_1114_mint__corr,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_mint(Ta) = some_nat(Xb) )
=> aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).
% mint_corr
tff(fact_1115_power__one,axiom,
! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).
% power_one
tff(fact_1116_power__one,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).
% power_one
tff(fact_1117_power__one,axiom,
! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).
% power_one
tff(fact_1118_power__one,axiom,
! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).
% power_one
tff(fact_1119_power__one,axiom,
! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).
% power_one
tff(fact_1120_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_1121_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_1122_power__one__right,axiom,
! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).
% power_one_right
tff(fact_1123_power__one__right,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).
% power_one_right
tff(fact_1124_semiring__norm_I78_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% semiring_norm(78)
tff(fact_1125_semiring__norm_I75_J,axiom,
! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).
% semiring_norm(75)
tff(fact_1126_power__inject__exp,axiom,
! [A: real,Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
<=> ( Mb = Nb ) ) ) ).
% power_inject_exp
tff(fact_1127_power__inject__exp,axiom,
! [A: rat,Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Nb) )
<=> ( Mb = Nb ) ) ) ).
% power_inject_exp
tff(fact_1128_power__inject__exp,axiom,
! [A: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
<=> ( Mb = Nb ) ) ) ).
% power_inject_exp
tff(fact_1129_power__inject__exp,axiom,
! [A: int,Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Nb) )
<=> ( Mb = Nb ) ) ) ).
% power_inject_exp
tff(fact_1130_semiring__norm_I76_J,axiom,
! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).
% semiring_norm(76)
tff(fact_1131_enat__ord__number_I2_J,axiom,
! [Mb: num,Nb: num] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).
% enat_ord_number(2)
tff(fact_1132_succ__member,axiom,
! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
<=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
& ! [Z3: nat] :
( ( aa_nat_o(vEBT_vebt_member(Ta),Z3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z3) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3) ) ) ) ).
% succ_member
tff(fact_1133_pred__member,axiom,
! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
<=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
& ! [Z3: nat] :
( ( aa_nat_o(vEBT_vebt_member(Ta),Z3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),Xb) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y) ) ) ) ).
% pred_member
tff(fact_1134_pred__corr,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Px) )
<=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Px) ) ) ).
% pred_corr
tff(fact_1135_succ__corr,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Sx) )
<=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Sx) ) ) ).
% succ_corr
tff(fact_1136_pred__correct,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Sx) )
<=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Ta)),Xb),Sx) ) ) ).
% pred_correct
tff(fact_1137_succ__correct,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Sx) )
<=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Ta)),Xb),Sx) ) ) ).
% succ_correct
tff(fact_1138_one__le__power,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).
% one_le_power
tff(fact_1139_one__le__power,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).
% one_le_power
tff(fact_1140_one__le__power,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ).
% one_le_power
tff(fact_1141_one__le__power,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).
% one_le_power
tff(fact_1142_power__gt1,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).
% power_gt1
tff(fact_1143_power__gt1,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).
% power_gt1
tff(fact_1144_power__gt1,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).
% power_gt1
tff(fact_1145_power__gt1,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))) ) ).
% power_gt1
tff(fact_1146_power__less__imp__less__exp,axiom,
! [A: real,Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% power_less_imp_less_exp
tff(fact_1147_power__less__imp__less__exp,axiom,
! [A: rat,Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% power_less_imp_less_exp
tff(fact_1148_power__less__imp__less__exp,axiom,
! [A: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% power_less_imp_less_exp
tff(fact_1149_power__less__imp__less__exp,axiom,
! [A: int,Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% power_less_imp_less_exp
tff(fact_1150_power__strict__increasing,axiom,
! [Nb: nat,N4: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).
% power_strict_increasing
tff(fact_1151_power__strict__increasing,axiom,
! [Nb: nat,N4: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).
% power_strict_increasing
tff(fact_1152_power__strict__increasing,axiom,
! [Nb: nat,N4: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).
% power_strict_increasing
tff(fact_1153_power__strict__increasing,axiom,
! [Nb: nat,N4: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).
% power_strict_increasing
tff(fact_1154_power__increasing,axiom,
! [Nb: nat,N4: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).
% power_increasing
tff(fact_1155_power__increasing,axiom,
! [Nb: nat,N4: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).
% power_increasing
tff(fact_1156_power__increasing,axiom,
! [Nb: nat,N4: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).
% power_increasing
tff(fact_1157_power__increasing,axiom,
! [Nb: nat,N4: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).
% power_increasing
tff(fact_1158_power__le__imp__le__exp,axiom,
! [A: rat,Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% power_le_imp_le_exp
tff(fact_1159_power__le__imp__le__exp,axiom,
! [A: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% power_le_imp_le_exp
tff(fact_1160_power__le__imp__le__exp,axiom,
! [A: int,Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% power_le_imp_le_exp
tff(fact_1161_power__le__imp__le__exp,axiom,
! [A: real,Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% power_le_imp_le_exp
tff(fact_1162_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_1163_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_1164_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_1165_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_1166_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_1167_less__exp,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% less_exp
tff(fact_1168_power2__nat__le__imp__le,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% power2_nat_le_imp_le
tff(fact_1169_power2__nat__le__eq__le,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% power2_nat_le_eq_le
tff(fact_1170_self__le__ge2__pow,axiom,
! [K: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)) ) ).
% self_le_ge2_pow
tff(fact_1171_two__powr__height__bound__deg,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(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,Ta))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).
% two_powr_height_bound_deg
tff(fact_1172__C5_Ohyps_C_I9_J,axiom,
( ( mi != ma )
=> ! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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),na) = I2 )
=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
& ! [X3: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),na) = I2 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),na)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),ma) ) ) ) ) ) ).
% "5.hyps"(9)
tff(fact_1173_vebt__maxt_Osimps_I3_J,axiom,
! [Mia: nat,Maa: 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(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Maa) ).
% vebt_maxt.simps(3)
tff(fact_1174_vebt__mint_Osimps_I3_J,axiom,
! [Mia: nat,Maa: 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(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Mia) ).
% vebt_mint.simps(3)
tff(fact_1175_geqmaxNone,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Maa),Xb)
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ).
% geqmaxNone
tff(fact_1176_greater__shift,axiom,
! [Y: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
<=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xb)),some_nat(Y)) ) ).
% greater_shift
tff(fact_1177_less__shift,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
<=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xb)),some_nat(Y)) ) ).
% less_shift
tff(fact_1178_nth__mem,axiom,
! [Nb: nat,Xs: list_P6011104703257516679at_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s5460976970255530739at_nat(Xs))
=> member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,Nb),set_Pr5648618587558075414at_nat(Xs)) ) ).
% nth_mem
tff(fact_1179_nth__mem,axiom,
! [Nb: nat,Xs: list_complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
=> aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).
% nth_mem
tff(fact_1180_nth__mem,axiom,
! [Nb: nat,Xs: list_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
=> aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).
% nth_mem
tff(fact_1181_nth__mem,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb)),set_VEBT_VEBT2(Xs)) ) ).
% nth_mem
tff(fact_1182_nth__mem,axiom,
! [Nb: nat,Xs: list_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
=> member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).
% nth_mem
tff(fact_1183_nth__mem,axiom,
! [Nb: nat,Xs: list_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
=> aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).
% nth_mem
tff(fact_1184_nth__mem,axiom,
! [Nb: nat,Xs: list_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
=> aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).
% nth_mem
tff(fact_1185_list__ball__nth,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs))
=> aa_VEBT_VEBT_o(P,X4) )
=> aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).
% list_ball_nth
tff(fact_1186_list__ball__nth,axiom,
! [Nb: nat,Xs: list_o,P: fun_o_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
=> ( ! [X4: $o] :
( member_o((X4),set_o2(Xs))
=> aa_o_o(P,(X4)) )
=> aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).
% list_ball_nth
tff(fact_1187_list__ball__nth,axiom,
! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
=> aa_nat_o(P,X4) )
=> aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).
% list_ball_nth
tff(fact_1188_list__ball__nth,axiom,
! [Nb: nat,Xs: list_int,P: fun_int_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),set_int2(Xs))
=> aa_int_o(P,X4) )
=> aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).
% list_ball_nth
tff(fact_1189_in__set__conv__nth,axiom,
! [Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs))
& ( nth_Pr7617993195940197384at_nat(Xs,I4) = Xb ) ) ) ).
% in_set_conv_nth
tff(fact_1190_in__set__conv__nth,axiom,
! [Xb: complex,Xs: list_complex] :
( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s3451745648224563538omplex(Xs))
& ( aa_nat_complex(nth_complex(Xs),I4) = Xb ) ) ) ).
% in_set_conv_nth
tff(fact_1191_in__set__conv__nth,axiom,
! [Xb: real,Xs: list_real] :
( aa_set_real_o(member_real(Xb),set_real2(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(Xs))
& ( aa_nat_real(nth_real(Xs),I4) = Xb ) ) ) ).
% in_set_conv_nth
tff(fact_1192_in__set__conv__nth,axiom,
! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
& ( nth_VEBT_VEBT(Xs,I4) = Xb ) ) ) ).
% in_set_conv_nth
tff(fact_1193_in__set__conv__nth,axiom,
! [Xb: $o,Xs: list_o] :
( member_o((Xb),set_o2(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
& ( aa_nat_o(nth_o(Xs),I4)
<=> (Xb) ) ) ) ).
% in_set_conv_nth
tff(fact_1194_in__set__conv__nth,axiom,
! [Xb: nat,Xs: list_nat] :
( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
& ( aa_nat_nat(nth_nat(Xs),I4) = Xb ) ) ) ).
% in_set_conv_nth
tff(fact_1195_in__set__conv__nth,axiom,
! [Xb: int,Xs: list_int] :
( aa_set_int_o(member_int(Xb),set_int2(Xs))
<=> ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
& ( aa_nat_int(nth_int(Xs),I4) = Xb ) ) ) ).
% in_set_conv_nth
tff(fact_1196_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Xb: product_prod_nat_nat] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
=> aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I3)) )
=> ( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
=> aa_Pro6150637604445964947_nat_o(P,Xb) ) ) ).
% all_nth_imp_all_set
tff(fact_1197_all__nth__imp__all__set,axiom,
! [Xs: list_complex,P: fun_complex_o,Xb: complex] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
=> aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I3)) )
=> ( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
=> aa_complex_o(P,Xb) ) ) ).
% all_nth_imp_all_set
tff(fact_1198_all__nth__imp__all__set,axiom,
! [Xs: list_real,P: fun_real_o,Xb: real] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
=> aa_real_o(P,aa_nat_real(nth_real(Xs),I3)) )
=> ( aa_set_real_o(member_real(Xb),set_real2(Xs))
=> aa_real_o(P,Xb) ) ) ).
% all_nth_imp_all_set
tff(fact_1199_all__nth__imp__all__set,axiom,
! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
=> aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) )
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
=> aa_VEBT_VEBT_o(P,Xb) ) ) ).
% all_nth_imp_all_set
tff(fact_1200_all__nth__imp__all__set,axiom,
! [Xs: list_o,P: fun_o_o,Xb: $o] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
=> aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
=> ( member_o((Xb),set_o2(Xs))
=> aa_o_o(P,(Xb)) ) ) ).
% all_nth_imp_all_set
tff(fact_1201_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
=> aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
=> ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
=> aa_nat_o(P,Xb) ) ) ).
% all_nth_imp_all_set
tff(fact_1202_all__nth__imp__all__set,axiom,
! [Xs: list_int,P: fun_int_o,Xb: int] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
=> aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
=> ( aa_set_int_o(member_int(Xb),set_int2(Xs))
=> aa_int_o(P,Xb) ) ) ).
% all_nth_imp_all_set
tff(fact_1203_bit__split__inv,axiom,
! [Xb: 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,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).
% bit_split_inv
tff(fact_1204_high__bound__aux,axiom,
! [Maa: nat,Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),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,Nb),Mb)))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).
% high_bound_aux
tff(fact_1205_high__inv,axiom,
! [Xb: nat,Nb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( 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))),Nb))),Xb)),Nb) = Y ) ) ).
% high_inv
tff(fact_1206_low__inv,axiom,
! [Xb: nat,Nb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( 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))),Nb))),Xb)),Nb) = Xb ) ) ).
% low_inv
tff(fact_1207_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_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_1208_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),times_7803423173614009249d_enat(numera1916890842035813515d_enat(W),Z)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_1209_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_1210_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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_1211_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_1212_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_1213_numeral__times__numeral,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% numeral_times_numeral
tff(fact_1214_numeral__times__numeral,axiom,
! [Mb: num,Nb: num] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% numeral_times_numeral
tff(fact_1215_numeral__times__numeral,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% numeral_times_numeral
tff(fact_1216_numeral__times__numeral,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% numeral_times_numeral
tff(fact_1217_numeral__times__numeral,axiom,
! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% numeral_times_numeral
tff(fact_1218_numeral__times__numeral,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% numeral_times_numeral
tff(fact_1219_mult__1,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).
% mult_1
tff(fact_1220_mult__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).
% mult_1
tff(fact_1221_mult__1,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).
% mult_1
tff(fact_1222_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_1223_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_1224_mult_Oright__neutral,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).
% mult.right_neutral
tff(fact_1225_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_1226_mult_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).
% mult.right_neutral
tff(fact_1227_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_1228_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_1229_not__Some__eq,axiom,
! [Xb: option4927543243414619207at_nat] :
( ! [Y5: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y5)
<=> ( Xb = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
tff(fact_1230_not__Some__eq,axiom,
! [Xb: option_nat] :
( ! [Y5: nat] : Xb != some_nat(Y5)
<=> ( Xb = none_nat ) ) ).
% not_Some_eq
tff(fact_1231_not__Some__eq,axiom,
! [Xb: option_num] :
( ! [Y5: num] : Xb != some_num(Y5)
<=> ( Xb = none_num ) ) ).
% not_Some_eq
tff(fact_1232_not__None__eq,axiom,
! [Xb: option4927543243414619207at_nat] :
( ( Xb != none_P5556105721700978146at_nat )
<=> ? [Y5: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y5) ) ).
% not_None_eq
tff(fact_1233_not__None__eq,axiom,
! [Xb: option_nat] :
( ( Xb != none_nat )
<=> ? [Y5: nat] : Xb = some_nat(Y5) ) ).
% not_None_eq
tff(fact_1234_not__None__eq,axiom,
! [Xb: option_num] :
( ( Xb != none_num )
<=> ? [Y5: num] : Xb = some_num(Y5) ) ).
% not_None_eq
tff(fact_1235_nat__mult__eq__1__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
<=> ( ( Mb = one_one_nat )
& ( Nb = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
tff(fact_1236_nat__1__eq__mult__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
<=> ( ( Mb = one_one_nat )
& ( Nb = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
tff(fact_1237_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_1238_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).
% distrib_right_numeral
tff(fact_1239_distrib__right__numeral,axiom,
! [A: extended_enat,B: extended_enat,V: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(A,B),numera1916890842035813515d_enat(V)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(V)),times_7803423173614009249d_enat(B,numera1916890842035813515d_enat(V))) ).
% distrib_right_numeral
tff(fact_1240_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_1241_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)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).
% distrib_right_numeral
tff(fact_1242_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_1243_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_1244_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).
% distrib_left_numeral
tff(fact_1245_distrib__left__numeral,axiom,
! [V: num,B: extended_enat,C: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(B,C)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),B),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),C)) ).
% distrib_left_numeral
tff(fact_1246_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_1247_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).
% distrib_left_numeral
tff(fact_1248_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_1249_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_1250_mult__Suc__right,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).
% mult_Suc_right
tff(fact_1251_power__add__numeral2,axiom,
! [A: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),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,Mb),Nb)))),B) ).
% power_add_numeral2
tff(fact_1252_power__add__numeral2,axiom,
! [A: real,Mb: num,Nb: 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,Mb))),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,Nb))),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,Mb),Nb)))),B) ).
% power_add_numeral2
tff(fact_1253_power__add__numeral2,axiom,
! [A: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),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,Mb),Nb)))),B) ).
% power_add_numeral2
tff(fact_1254_power__add__numeral2,axiom,
! [A: nat,Mb: num,Nb: 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,Mb))),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,Nb))),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,Mb),Nb)))),B) ).
% power_add_numeral2
tff(fact_1255_power__add__numeral2,axiom,
! [A: int,Mb: num,Nb: 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,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),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,Mb),Nb)))),B) ).
% power_add_numeral2
tff(fact_1256_power__add__numeral,axiom,
! [A: complex,Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% power_add_numeral
tff(fact_1257_power__add__numeral,axiom,
! [A: real,Mb: num,Nb: 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,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = 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,Mb),Nb))) ).
% power_add_numeral
tff(fact_1258_power__add__numeral,axiom,
! [A: rat,Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% power_add_numeral
tff(fact_1259_power__add__numeral,axiom,
! [A: nat,Mb: num,Nb: 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,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = 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,Mb),Nb))) ).
% power_add_numeral
tff(fact_1260_power__add__numeral,axiom,
! [A: int,Mb: num,Nb: 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,Mb))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% power_add_numeral
tff(fact_1261_enat__less__induct,axiom,
! [P: fun_Extended_enat_o,Nb: extended_enat] :
( ! [N: extended_enat] :
( ! [M3: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(M3),N)
=> aa_Extended_enat_o(P,M3) )
=> aa_Extended_enat_o(P,N) )
=> aa_Extended_enat_o(P,Nb) ) ).
% enat_less_induct
tff(fact_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_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_1274_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_1275_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_1276_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_1277_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_1278_VEBT__internal_Ooption__shift_Ocases,axiom,
! [Xb: produc1260151622200656695at_nat] :
( ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : Xb != produc7814092408141220007at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
=> ( ! [Uw: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
=> ~ ! [F2: fun_Pr2477239776880934807at_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B4))) ) ) ).
% VEBT_internal.option_shift.cases
tff(fact_1279_VEBT__internal_Ooption__shift_Ocases,axiom,
! [Xb: produc7004362771917097900on_nat] :
( ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : Xb != produc6699827698519632094on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
=> ( ! [Uw: fun_nat_fun_nat_nat,V2: nat] : Xb != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(some_nat(V2),none_nat))
=> ~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B4: nat] : Xb != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A4),some_nat(B4))) ) ) ).
% VEBT_internal.option_shift.cases
tff(fact_1280_VEBT__internal_Ooption__shift_Ocases,axiom,
! [Xb: produc2703417854250991842on_num] :
( ! [Uu: fun_num_fun_num_num,Uv: option_num] : Xb != produc6360833703471310868on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
=> ( ! [Uw: fun_num_fun_num_num,V2: num] : Xb != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(some_num(V2),none_num))
=> ~ ! [F2: fun_num_fun_num_num,A4: num,B4: num] : Xb != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A4),some_num(B4))) ) ) ).
% VEBT_internal.option_shift.cases
tff(fact_1281_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [Xb: produc3739950874144486240at_nat] :
( ! [Uu: fun_Pr2028404973800837934_nat_o,Uv: option4927543243414619207at_nat] : Xb != produc3653971535690468304at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
=> ( ! [Uw: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
=> ~ ! [F2: fun_Pr2028404973800837934_nat_o,X4: product_prod_nat_nat,Y3: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X4),some_P7363390416028606310at_nat(Y3))) ) ) ).
% VEBT_internal.option_comp_shift.cases
tff(fact_1282_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [Xb: produc1615844007061411358on_nat] :
( ! [Uu: fun_nat_fun_nat_o,Uv: option_nat] : Xb != produc7967433646104525454on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
=> ( ! [Uw: fun_nat_fun_nat_o,V2: nat] : Xb != produc7967433646104525454on_nat(Uw,produc5098337634421038937on_nat(some_nat(V2),none_nat))
=> ~ ! [F2: fun_nat_fun_nat_o,X4: nat,Y3: nat] : Xb != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X4),some_nat(Y3))) ) ) ).
% VEBT_internal.option_comp_shift.cases
tff(fact_1283_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [Xb: produc50614890452615238on_num] :
( ! [Uu: fun_num_fun_num_o,Uv: option_num] : Xb != produc1140783415258738358on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
=> ( ! [Uw: fun_num_fun_num_o,V2: num] : Xb != produc1140783415258738358on_num(Uw,produc8585076106096196333on_num(some_num(V2),none_num))
=> ~ ! [F2: fun_num_fun_num_o,X4: num,Y3: num] : Xb != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X4),some_num(Y3))) ) ) ).
% VEBT_internal.option_comp_shift.cases
tff(fact_1284_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu2,none_P5556105721700978146at_nat,Uv2) = none_P5556105721700978146at_nat ).
% VEBT_internal.option_shift.simps(1)
tff(fact_1285_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu2: fun_num_fun_num_num,Uv2: option_num] : vEBT_V819420779217536731ft_num(Uu2,none_num,Uv2) = none_num ).
% VEBT_internal.option_shift.simps(1)
tff(fact_1286_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu2),none_nat),Uv2) = none_nat ).
% VEBT_internal.option_shift.simps(1)
tff(fact_1287_mult_Ocomm__neutral,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).
% mult.comm_neutral
tff(fact_1288_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_1289_mult_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).
% mult.comm_neutral
tff(fact_1290_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_1291_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_1292_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_1293_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_1294_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_1295_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_1296_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_1297_combine__options__cases,axiom,
! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
( ( ( Xb = none_P5556105721700978146at_nat )
=> aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
=> ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
( ( Xb = some_P7363390416028606310at_nat(A4) )
=> ( ( Y = some_P7363390416028606310at_nat(B4) )
=> aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) )
=> aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1298_combine__options__cases,axiom,
! [Xb: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
( ( ( Xb = none_P5556105721700978146at_nat )
=> aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
=> ( ( ( Y = none_nat )
=> aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
=> ( ! [A4: product_prod_nat_nat,B4: nat] :
( ( Xb = some_P7363390416028606310at_nat(A4) )
=> ( ( Y = some_nat(B4) )
=> aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) )
=> aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1299_combine__options__cases,axiom,
! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
( ( ( Xb = none_P5556105721700978146at_nat )
=> aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
=> ( ( ( Y = none_num )
=> aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
=> ( ! [A4: product_prod_nat_nat,B4: num] :
( ( Xb = some_P7363390416028606310at_nat(A4) )
=> ( ( Y = some_num(B4) )
=> aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) )
=> aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1300_combine__options__cases,axiom,
! [Xb: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
( ( ( Xb = none_nat )
=> aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
=> ( ! [A4: nat,B4: product_prod_nat_nat] :
( ( Xb = some_nat(A4) )
=> ( ( Y = some_P7363390416028606310at_nat(B4) )
=> aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) )
=> aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1301_combine__options__cases,axiom,
! [Xb: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
( ( ( Xb = none_nat )
=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
=> ( ( ( Y = none_nat )
=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
=> ( ! [A4: nat,B4: nat] :
( ( Xb = some_nat(A4) )
=> ( ( Y = some_nat(B4) )
=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) )
=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1302_combine__options__cases,axiom,
! [Xb: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
( ( ( Xb = none_nat )
=> aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
=> ( ( ( Y = none_num )
=> aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
=> ( ! [A4: nat,B4: num] :
( ( Xb = some_nat(A4) )
=> ( ( Y = some_num(B4) )
=> aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) )
=> aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1303_combine__options__cases,axiom,
! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
( ( ( Xb = none_num )
=> aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
=> ( ! [A4: num,B4: product_prod_nat_nat] :
( ( Xb = some_num(A4) )
=> ( ( Y = some_P7363390416028606310at_nat(B4) )
=> aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) )
=> aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1304_combine__options__cases,axiom,
! [Xb: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
( ( ( Xb = none_num )
=> aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
=> ( ( ( Y = none_nat )
=> aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
=> ( ! [A4: num,B4: nat] :
( ( Xb = some_num(A4) )
=> ( ( Y = some_nat(B4) )
=> aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) )
=> aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1305_combine__options__cases,axiom,
! [Xb: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
( ( ( Xb = none_num )
=> aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
=> ( ( ( Y = none_num )
=> aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
=> ( ! [A4: num,B4: num] :
( ( Xb = some_num(A4) )
=> ( ( Y = some_num(B4) )
=> aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) )
=> aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) ) ) ).
% combine_options_cases
tff(fact_1306_split__option__all,axiom,
! [P: fun_op8028897214545316308_nat_o] :
( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
<=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
& ! [X: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X)) ) ) ).
% split_option_all
tff(fact_1307_split__option__all,axiom,
! [P: fun_option_nat_o] :
( ! [X_12: option_nat] : aa_option_nat_o(P,X_12)
<=> ( aa_option_nat_o(P,none_nat)
& ! [X: nat] : aa_option_nat_o(P,some_nat(X)) ) ) ).
% split_option_all
tff(fact_1308_split__option__all,axiom,
! [P: fun_option_num_o] :
( ! [X_12: option_num] : aa_option_num_o(P,X_12)
<=> ( aa_option_num_o(P,none_num)
& ! [X: num] : aa_option_num_o(P,some_num(X)) ) ) ).
% split_option_all
tff(fact_1309_split__option__ex,axiom,
! [P: fun_op8028897214545316308_nat_o] :
( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
<=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
| ? [X: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X)) ) ) ).
% split_option_ex
tff(fact_1310_split__option__ex,axiom,
! [P: fun_option_nat_o] :
( ? [X_12: option_nat] : aa_option_nat_o(P,X_12)
<=> ( aa_option_nat_o(P,none_nat)
| ? [X: nat] : aa_option_nat_o(P,some_nat(X)) ) ) ).
% split_option_ex
tff(fact_1311_split__option__ex,axiom,
! [P: fun_option_num_o] :
( ? [X_12: option_num] : aa_option_num_o(P,X_12)
<=> ( aa_option_num_o(P,none_num)
| ? [X: num] : aa_option_num_o(P,some_num(X)) ) ) ).
% split_option_ex
tff(fact_1312_option_Oexhaust,axiom,
! [Y: option4927543243414619207at_nat] :
( ( Y != none_P5556105721700978146at_nat )
=> ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).
% option.exhaust
tff(fact_1313_option_Oexhaust,axiom,
! [Y: option_nat] :
( ( Y != none_nat )
=> ~ ! [X22: nat] : Y != some_nat(X22) ) ).
% option.exhaust
tff(fact_1314_option_Oexhaust,axiom,
! [Y: option_num] :
( ( Y != none_num )
=> ~ ! [X22: num] : Y != some_num(X22) ) ).
% option.exhaust
tff(fact_1315_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
( ( Option = some_P7363390416028606310at_nat(X2) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
tff(fact_1316_option_OdiscI,axiom,
! [Option: option_nat,X2: nat] :
( ( Option = some_nat(X2) )
=> ( Option != none_nat ) ) ).
% option.discI
tff(fact_1317_option_OdiscI,axiom,
! [Option: option_num,X2: num] :
( ( Option = some_num(X2) )
=> ( Option != none_num ) ) ).
% option.discI
tff(fact_1318_option_Odistinct_I1_J,axiom,
! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).
% option.distinct(1)
tff(fact_1319_option_Odistinct_I1_J,axiom,
! [X2: nat] : none_nat != some_nat(X2) ).
% option.distinct(1)
tff(fact_1320_option_Odistinct_I1_J,axiom,
! [X2: num] : none_num != some_num(X2) ).
% option.distinct(1)
tff(fact_1321_power__commutes,axiom,
! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).
% power_commutes
tff(fact_1322_power__commutes,axiom,
! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% power_commutes
tff(fact_1323_power__commutes,axiom,
! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).
% power_commutes
tff(fact_1324_power__commutes,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),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),Nb)) ).
% power_commutes
tff(fact_1325_power__commutes,axiom,
! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).
% power_commutes
tff(fact_1326_power__mult__distrib,axiom,
! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).
% power_mult_distrib
tff(fact_1327_power__mult__distrib,axiom,
! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).
% power_mult_distrib
tff(fact_1328_power__mult__distrib,axiom,
! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).
% power_mult_distrib
tff(fact_1329_power__mult__distrib,axiom,
! [A: nat,B: nat,Nb: 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)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).
% power_mult_distrib
tff(fact_1330_power__mult__distrib,axiom,
! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ).
% power_mult_distrib
tff(fact_1331_power__commuting__commutes,axiom,
! [Xb: complex,Y: complex,Nb: nat] :
( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(Xb),Nb)) ) ) ).
% power_commuting_commutes
tff(fact_1332_power__commuting__commutes,axiom,
! [Xb: real,Y: real,Nb: nat] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).
% power_commuting_commutes
tff(fact_1333_power__commuting__commutes,axiom,
! [Xb: rat,Y: rat,Nb: nat] :
( ( aa_rat_rat(times_times_rat(Xb),Y) = aa_rat_rat(times_times_rat(Y),Xb) )
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).
% power_commuting_commutes
tff(fact_1334_power__commuting__commutes,axiom,
! [Xb: nat,Y: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xb) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).
% power_commuting_commutes
tff(fact_1335_power__commuting__commutes,axiom,
! [Xb: int,Y: int,Nb: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb) )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(Xb),Nb)) ) ) ).
% power_commuting_commutes
tff(fact_1336_Suc__mult__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
<=> ( Mb = Nb ) ) ).
% Suc_mult_cancel1
tff(fact_1337_power__mult,axiom,
! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),Nb) ).
% power_mult
tff(fact_1338_power__mult,axiom,
! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),Nb) ).
% power_mult
tff(fact_1339_power__mult,axiom,
! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Mb)),Nb) ).
% power_mult
tff(fact_1340_power__mult,axiom,
! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Mb)),Nb) ).
% power_mult
tff(fact_1341_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).
% mult_le_mono2
tff(fact_1342_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).
% mult_le_mono1
tff(fact_1343_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).
% mult_le_mono
tff(fact_1344_le__square,axiom,
! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).
% le_square
tff(fact_1345_le__cube,axiom,
! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).
% le_cube
tff(fact_1346_add__mult__distrib2,axiom,
! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).
% add_mult_distrib2
tff(fact_1347_add__mult__distrib,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).
% add_mult_distrib
tff(fact_1348_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_1349_nat__mult__1__right,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).
% nat_mult_1_right
tff(fact_1350_nat__mult__1,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).
% nat_mult_1
tff(fact_1351_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw2: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw2,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).
% VEBT_internal.option_shift.simps(2)
tff(fact_1352_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw2: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw2,some_num(V),none_num) = none_num ).
% VEBT_internal.option_shift.simps(2)
tff(fact_1353_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw2: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw2),some_nat(V)),none_nat) = none_nat ).
% VEBT_internal.option_shift.simps(2)
tff(fact_1354_VEBT__internal_Ooption__shift_Oelims,axiom,
! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
=> ( ( ( Xaa = none_P5556105721700978146at_nat )
=> ( Y != none_P5556105721700978146at_nat ) )
=> ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
=> ( ( Xba = none_P5556105721700978146at_nat )
=> ( Y != none_P5556105721700978146at_nat ) ) )
=> ~ ! [A4: product_prod_nat_nat] :
( ( Xaa = some_P7363390416028606310at_nat(A4) )
=> ! [B4: product_prod_nat_nat] :
( ( Xba = some_P7363390416028606310at_nat(B4) )
=> ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A4),B4)) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
tff(fact_1355_VEBT__internal_Ooption__shift_Oelims,axiom,
! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y )
=> ( ( ( Xaa = none_num )
=> ( Y != none_num ) )
=> ( ( ? [V2: num] : Xaa = some_num(V2)
=> ( ( Xba = none_num )
=> ( Y != none_num ) ) )
=> ~ ! [A4: num] :
( ( Xaa = some_num(A4) )
=> ! [B4: num] :
( ( Xba = some_num(B4) )
=> ( Y != some_num(aa_num_num(aa_num_fun_num_num(Xb,A4),B4)) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
tff(fact_1356_VEBT__internal_Ooption__shift_Oelims,axiom,
! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
=> ( ( ( Xaa = none_nat )
=> ( Y != none_nat ) )
=> ( ( ? [V2: nat] : Xaa = some_nat(V2)
=> ( ( Xba = none_nat )
=> ( Y != none_nat ) ) )
=> ~ ! [A4: nat] :
( ( Xaa = some_nat(A4) )
=> ! [B4: nat] :
( ( Xba = some_nat(B4) )
=> ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A4),B4)) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
tff(fact_1357_mult__numeral__1,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).
% mult_numeral_1
tff(fact_1358_mult__numeral__1,axiom,
! [A: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(one),A) = A ).
% mult_numeral_1
tff(fact_1359_mult__numeral__1,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).
% mult_numeral_1
tff(fact_1360_mult__numeral__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A) = A ).
% mult_numeral_1
tff(fact_1361_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_1362_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_1363_mult__numeral__1__right,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).
% mult_numeral_1_right
tff(fact_1364_mult__numeral__1__right,axiom,
! [A: extended_enat] : times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(one)) = A ).
% mult_numeral_1_right
tff(fact_1365_mult__numeral__1__right,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).
% mult_numeral_1_right
tff(fact_1366_mult__numeral__1__right,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,one)) = A ).
% mult_numeral_1_right
tff(fact_1367_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_1368_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_1369_left__right__inverse__power,axiom,
! [Xb: complex,Y: complex,Nb: nat] :
( ( aa_complex_complex(times_times_complex(Xb),Y) = one_one_complex )
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = one_one_complex ) ) ).
% left_right_inverse_power
tff(fact_1370_left__right__inverse__power,axiom,
! [Xb: real,Y: real,Nb: nat] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).
% left_right_inverse_power
tff(fact_1371_left__right__inverse__power,axiom,
! [Xb: rat,Y: rat,Nb: nat] :
( ( aa_rat_rat(times_times_rat(Xb),Y) = one_one_rat )
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).
% left_right_inverse_power
tff(fact_1372_left__right__inverse__power,axiom,
! [Xb: nat,Y: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),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,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).
% left_right_inverse_power
tff(fact_1373_left__right__inverse__power,axiom,
! [Xb: int,Y: int,Nb: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = one_one_int )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) = one_one_int ) ) ).
% left_right_inverse_power
tff(fact_1374_power__Suc2,axiom,
! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).
% power_Suc2
tff(fact_1375_power__Suc2,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).
% power_Suc2
tff(fact_1376_power__Suc2,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).
% power_Suc2
tff(fact_1377_power__Suc2,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).
% power_Suc2
tff(fact_1378_power__Suc2,axiom,
! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).
% power_Suc2
tff(fact_1379_power__Suc,axiom,
! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).
% power_Suc
tff(fact_1380_power__Suc,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% power_Suc
tff(fact_1381_power__Suc,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).
% power_Suc
tff(fact_1382_power__Suc,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).
% power_Suc
tff(fact_1383_power__Suc,axiom,
! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).
% power_Suc
tff(fact_1384_Suc__mult__less__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% Suc_mult_less_cancel1
tff(fact_1385_power__add,axiom,
! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).
% power_add
tff(fact_1386_power__add,axiom,
! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% power_add
tff(fact_1387_power__add,axiom,
! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).
% power_add
tff(fact_1388_power__add,axiom,
! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).
% power_add
tff(fact_1389_power__add,axiom,
! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).
% power_add
tff(fact_1390_Suc__mult__le__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% Suc_mult_le_cancel1
tff(fact_1391_mult__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).
% mult_Suc
tff(fact_1392_power__odd__eq,axiom,
! [A: complex,Nb: 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))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1393_power__odd__eq,axiom,
! [A: real,Nb: 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))),Nb))) = 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),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1394_power__odd__eq,axiom,
! [A: rat,Nb: 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))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1395_power__odd__eq,axiom,
! [A: nat,Nb: 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))),Nb))) = 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),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1396_power__odd__eq,axiom,
! [A: int,Nb: 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))),Nb))) = 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),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1397_power__less__power__Suc,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).
% power_less_power_Suc
tff(fact_1398_power__less__power__Suc,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).
% power_less_power_Suc
tff(fact_1399_power__less__power__Suc,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).
% power_less_power_Suc
tff(fact_1400_power__less__power__Suc,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).
% power_less_power_Suc
tff(fact_1401_power__gt1__lemma,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> aa_real_o(aa_real_fun_real_o(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),Nb))) ) ).
% power_gt1_lemma
tff(fact_1402_power__gt1__lemma,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).
% power_gt1_lemma
tff(fact_1403_power__gt1__lemma,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(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),Nb))) ) ).
% power_gt1_lemma
tff(fact_1404_power__gt1__lemma,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(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),Nb))) ) ).
% power_gt1_lemma
tff(fact_1405_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B3: set_Pr1261947904930325089at_nat] :
( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),B3)
<=> ! [X: product_prod_nat_nat] :
( member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(Xs))
=> member8440522571783428010at_nat(X,B3) ) ) ).
% subset_code(1)
tff(fact_1406_subset__code_I1_J,axiom,
! [Xs: list_complex,B3: set_complex] :
( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B3)
<=> ! [X: complex] :
( aa_set_complex_o(member_complex(X),set_complex2(Xs))
=> aa_set_complex_o(member_complex(X),B3) ) ) ).
% subset_code(1)
tff(fact_1407_subset__code_I1_J,axiom,
! [Xs: list_real,B3: set_real] :
( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B3)
<=> ! [X: real] :
( aa_set_real_o(member_real(X),set_real2(Xs))
=> aa_set_real_o(member_real(X),B3) ) ) ).
% subset_code(1)
tff(fact_1408_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B3)
<=> ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B3) ) ) ).
% subset_code(1)
tff(fact_1409_subset__code_I1_J,axiom,
! [Xs: list_int,B3: set_int] :
( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B3)
<=> ! [X: int] :
( aa_set_int_o(member_int(X),set_int2(Xs))
=> aa_set_int_o(member_int(X),B3) ) ) ).
% subset_code(1)
tff(fact_1410_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B3)
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),set_nat2(Xs))
=> aa_set_nat_o(member_nat(X),B3) ) ) ).
% subset_code(1)
tff(fact_1411_Ex__list__of__length,axiom,
! [Nb: nat] :
? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).
% Ex_list_of_length
tff(fact_1412_Ex__list__of__length,axiom,
! [Nb: nat] :
? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).
% Ex_list_of_length
tff(fact_1413_Ex__list__of__length,axiom,
! [Nb: nat] :
? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).
% Ex_list_of_length
tff(fact_1414_Ex__list__of__length,axiom,
! [Nb: nat] :
? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).
% Ex_list_of_length
tff(fact_1415_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_1416_neq__if__length__neq,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_1417_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_1418_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_1419_mult__2,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).
% mult_2
tff(fact_1420_mult__2,axiom,
! [Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Z) = plus_p3455044024723400733d_enat(Z,Z) ).
% mult_2
tff(fact_1421_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_1422_mult__2,axiom,
! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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_1423_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_1424_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_1425_mult__2__right,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).
% mult_2_right
tff(fact_1426_mult__2__right,axiom,
! [Z: extended_enat] : times_7803423173614009249d_enat(Z,numera1916890842035813515d_enat(aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z,Z) ).
% mult_2_right
tff(fact_1427_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_1428_mult__2__right,axiom,
! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).
% mult_2_right
tff(fact_1429_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_1430_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_1431_left__add__twice,axiom,
! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_1432_left__add__twice,axiom,
! [A: extended_enat,B: extended_enat] : plus_p3455044024723400733d_enat(A,plus_p3455044024723400733d_enat(A,B)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),B) ).
% left_add_twice
tff(fact_1433_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_1434_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_1435_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_1436_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_1437_power4__eq__xxxx,axiom,
! [Xb: complex] : aa_nat_complex(power_power_complex(Xb),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(Xb),Xb)),Xb)),Xb) ).
% power4_eq_xxxx
tff(fact_1438_power4__eq__xxxx,axiom,
! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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,Xb),Xb)),Xb)),Xb) ).
% power4_eq_xxxx
tff(fact_1439_power4__eq__xxxx,axiom,
! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),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(Xb),Xb)),Xb)),Xb) ).
% power4_eq_xxxx
tff(fact_1440_power4__eq__xxxx,axiom,
! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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,Xb),Xb)),Xb)),Xb) ).
% power4_eq_xxxx
tff(fact_1441_power4__eq__xxxx,axiom,
! [Xb: int] : aa_nat_int(power_power_int(Xb),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,Xb),Xb)),Xb)),Xb) ).
% power4_eq_xxxx
tff(fact_1442_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_1443_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_1444_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_1445_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_1446_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_1447_power__even__eq,axiom,
! [A: nat,Nb: 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))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_1448_power__even__eq,axiom,
! [A: real,Nb: 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))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_1449_power__even__eq,axiom,
! [A: int,Nb: 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))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_1450_power__even__eq,axiom,
! [A: complex,Nb: 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))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_1451_power2__sum,axiom,
! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)) ).
% power2_sum
tff(fact_1452_power2__sum,axiom,
! [Xb: extended_enat,Y: extended_enat] : power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(Xb,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(power_8040749407984259932d_enat(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8040749407984259932d_enat(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_7803423173614009249d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Xb),Y)) ).
% power2_sum
tff(fact_1453_power2__sum,axiom,
! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),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(Xb),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))),Xb)),Y)) ).
% power2_sum
tff(fact_1454_power2__sum,axiom,
! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)) ).
% power2_sum
tff(fact_1455_power2__sum,axiom,
! [Xb: 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,Xb),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,Xb),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))),Xb)),Y)) ).
% power2_sum
tff(fact_1456_power2__sum,axiom,
! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),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(Xb),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))),Xb)),Y)) ).
% power2_sum
tff(fact_1457_length__induct,axiom,
! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
( ! [Xs2: list_VEBT_VEBT] :
( ! [Ys2: list_VEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
=> aa_list_VEBT_VEBT_o(P,Ys2) )
=> aa_list_VEBT_VEBT_o(P,Xs2) )
=> aa_list_VEBT_VEBT_o(P,Xs) ) ).
% length_induct
tff(fact_1458_length__induct,axiom,
! [P: fun_list_o_o,Xs: list_o] :
( ! [Xs2: list_o] :
( ! [Ys2: list_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
=> aa_list_o_o(P,Ys2) )
=> aa_list_o_o(P,Xs2) )
=> aa_list_o_o(P,Xs) ) ).
% length_induct
tff(fact_1459_length__induct,axiom,
! [P: fun_list_nat_o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
=> aa_list_nat_o(P,Ys2) )
=> aa_list_nat_o(P,Xs2) )
=> aa_list_nat_o(P,Xs) ) ).
% length_induct
tff(fact_1460_length__induct,axiom,
! [P: fun_list_int_o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
=> aa_list_int_o(P,Ys2) )
=> aa_list_int_o(P,Xs2) )
=> aa_list_int_o(P,Xs) ) ).
% length_induct
tff(fact_1461_invar__vebt_Ointros_I4_J,axiom,
! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(vEBT_invar_vebt(X4),Nb) )
=> ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
=> ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
=> ( ( Mb = Nb )
=> ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
=> ( ( ( Mia = Maa )
=> ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
=> ( ( ( Mia != Maa )
=> ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
& ! [X4: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I3 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X4)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
=> aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
tff(fact_1462_invar__vebt_Ointros_I5_J,axiom,
! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(vEBT_invar_vebt(X4),Nb) )
=> ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
=> ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
=> ( ( Mb = aa_nat_nat(suc,Nb) )
=> ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
=> ( ( ( Mia = Maa )
=> ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
=> ( ( ( Mia != Maa )
=> ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
& ! [X4: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I3 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X4)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
=> aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
tff(fact_1463_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_1464_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_1465_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_1466_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
=> ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1467_nth__equalityI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
=> ( aa_nat_o(nth_o(Xs),I3)
<=> aa_nat_o(nth_o(Ys),I3) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1468_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
=> ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1469_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
=> ( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
=> ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1470_Skolem__list__nth,axiom,
! [K: nat,P: fun_na6887940067898170868VEBT_o] :
( ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),X_12) )
<=> ? [Xs3: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs3) = K )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),nth_VEBT_VEBT(Xs3,I4)) ) ) ) ).
% Skolem_list_nth
tff(fact_1471_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_o_o] :
( ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I4),(X_12)) )
<=> ? [Xs3: list_o] :
( ( size_size_list_o(Xs3) = K )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> aa_o_o(aa_nat_fun_o_o(P,I4),aa_nat_o(nth_o(Xs3),I4)) ) ) ) ).
% Skolem_list_nth
tff(fact_1472_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_nat_o] :
( ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I4),X_12) )
<=> ? [Xs3: list_nat] :
( ( size_size_list_nat(Xs3) = K )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> aa_nat_o(aa_nat_fun_nat_o(P,I4),aa_nat_nat(nth_nat(Xs3),I4)) ) ) ) ).
% Skolem_list_nth
tff(fact_1473_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_int_o] :
( ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I4),X_12) )
<=> ? [Xs3: list_int] :
( ( size_size_list_int(Xs3) = K )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
=> aa_int_o(aa_nat_fun_int_o(P,I4),aa_nat_int(nth_int(Xs3),I4)) ) ) ) ).
% Skolem_list_nth
tff(fact_1474_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] :
( aa_nat_o(aa_nat_fun_nat_o(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_1475_list__eq__iff__nth__eq,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs = Ys )
<=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
=> ( aa_nat_o(nth_o(Xs),I4)
<=> aa_nat_o(nth_o(Ys),I4) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1476_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] :
( aa_nat_o(aa_nat_fun_nat_o(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_1477_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] :
( aa_nat_o(aa_nat_fun_nat_o(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_1478_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
=> aa_VEBT_VEBT_o(P,X) )
<=> ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
=> aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I4)) ) ) ).
% all_set_conv_all_nth
tff(fact_1479_all__set__conv__all__nth,axiom,
! [Xs: list_o,P: fun_o_o] :
( ! [X: $o] :
( member_o((X),set_o2(Xs))
=> aa_o_o(P,(X)) )
<=> ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
=> aa_o_o(P,aa_nat_o(nth_o(Xs),I4)) ) ) ).
% all_set_conv_all_nth
tff(fact_1480_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: fun_nat_o] :
( ! [X: nat] :
( aa_set_nat_o(member_nat(X),set_nat2(Xs))
=> aa_nat_o(P,X) )
<=> ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
=> aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I4)) ) ) ).
% all_set_conv_all_nth
tff(fact_1481_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: fun_int_o] :
( ! [X: int] :
( aa_set_int_o(member_int(X),set_int2(Xs))
=> aa_int_o(P,X) )
<=> ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
=> aa_int_o(P,aa_nat_int(nth_int(Xs),I4)) ) ) ).
% all_set_conv_all_nth
tff(fact_1482_in__children__def,axiom,
! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).
% in_children_def
tff(fact_1483_both__member__options__from__chilf__to__complete__tree,axiom,
! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
tff(fact_1484_member__inv,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
& ( ( Xb = Mia )
| ( Xb = Maa )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
& aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).
% member_inv
tff(fact_1485_both__member__options__from__complete__tree__to__child,axiom,
! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
| ( Xb = Mia )
| ( Xb = Maa ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
tff(fact_1486_succ__list__to__short,axiom,
! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).
% succ_list_to_short
tff(fact_1487_pred__list__to__short,axiom,
! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).
% pred_list_to_short
tff(fact_1488_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
=> ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).
% both_member_options_ding
tff(fact_1489_sum__squares__bound,axiom,
! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_1490_sum__squares__bound,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_1491_mul__def,axiom,
vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).
% mul_def
tff(fact_1492_Suc__double__not__eq__double,axiom,
! [Mb: nat,Nb: 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))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).
% Suc_double_not_eq_double
tff(fact_1493_double__not__eq__Suc__double,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != 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))),Nb)) ).
% double_not_eq_Suc_double
tff(fact_1494_mul__shift,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = Z )
<=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).
% mul_shift
tff(fact_1495_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_1496_high__def,axiom,
! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% high_def
tff(fact_1497_semiring__norm_I13_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% semiring_norm(13)
tff(fact_1498_semiring__norm_I11_J,axiom,
! [Mb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Mb),one) = Mb ).
% semiring_norm(11)
tff(fact_1499_semiring__norm_I12_J,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).
% semiring_norm(12)
tff(fact_1500_num__double,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).
% num_double
tff(fact_1501_power__mult__numeral,axiom,
! [A: nat,Mb: num,Nb: 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,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = 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,Mb),Nb))) ).
% power_mult_numeral
tff(fact_1502_power__mult__numeral,axiom,
! [A: real,Mb: num,Nb: 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,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = 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,Mb),Nb))) ).
% power_mult_numeral
tff(fact_1503_power__mult__numeral,axiom,
! [A: int,Mb: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% power_mult_numeral
tff(fact_1504_power__mult__numeral,axiom,
! [A: complex,Mb: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% power_mult_numeral
tff(fact_1505_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).
% le_divide_eq_numeral1(1)
tff(fact_1506_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B) ) ).
% le_divide_eq_numeral1(1)
tff(fact_1507_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).
% divide_le_eq_numeral1(1)
tff(fact_1508_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W))),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).
% divide_le_eq_numeral1(1)
tff(fact_1509_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B) ) ).
% less_divide_eq_numeral1(1)
tff(fact_1510_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).
% less_divide_eq_numeral1(1)
tff(fact_1511_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W))),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).
% divide_less_eq_numeral1(1)
tff(fact_1512_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).
% divide_less_eq_numeral1(1)
tff(fact_1513_power__divide,axiom,
! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).
% power_divide
tff(fact_1514_power__divide,axiom,
! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).
% power_divide
tff(fact_1515_power__divide,axiom,
! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).
% power_divide
tff(fact_1516_divide__numeral__1,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).
% divide_numeral_1
tff(fact_1517_divide__numeral__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,one)) = A ).
% divide_numeral_1
tff(fact_1518_divide__numeral__1,axiom,
! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).
% divide_numeral_1
tff(fact_1519_power__one__over,axiom,
! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),Nb)) ).
% power_one_over
tff(fact_1520_power__one__over,axiom,
! [A: real,Nb: 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)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% power_one_over
tff(fact_1521_power__one__over,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ).
% power_one_over
tff(fact_1522_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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_1523_four__x__squared,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% four_x_squared
tff(fact_1524_vebt__mint_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).
% vebt_mint.simps(2)
tff(fact_1525_vebt__maxt_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).
% vebt_maxt.simps(2)
tff(fact_1526_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va)),Vb) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_1527_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,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),Xb) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_1528_invar__vebt_Ointros_I2_J,axiom,
! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(vEBT_invar_vebt(X4),Nb) )
=> ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
=> ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
=> ( ( Mb = Nb )
=> ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
=> ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
=> aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
tff(fact_1529_invar__vebt_Ointros_I3_J,axiom,
! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(vEBT_invar_vebt(X4),Nb) )
=> ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
=> ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
=> ( ( Mb = aa_nat_nat(suc,Nb) )
=> ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
=> ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
=> ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
=> aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
tff(fact_1530_nested__mint,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Va: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
=> ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Mia)
=> ( ( Maa != Mia )
=> aa_nat_o(aa_nat_fun_nat_o(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(Summarya))),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(TreeLista,the_nat(vEBT_vebt_mint(Summarya))))))),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(TreeLista)) ) ) ) ) ).
% nested_mint
tff(fact_1531_add__self__div__2,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).
% add_self_div_2
tff(fact_1532_div2__Suc__Suc,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% div2_Suc_Suc
tff(fact_1533_mintlistlength,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
=> ( ( Mia != Maa )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
& ? [M2: nat] :
( ( some_nat(M2) = vEBT_vebt_mint(Summarya) )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).
% mintlistlength
tff(fact_1534_summaxma,axiom,
! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
=> ( ( Mia != Maa )
=> ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% summaxma
tff(fact_1535_div__exp__eq,axiom,
! [A: nat,Mb: nat,Nb: 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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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,Mb),Nb))) ).
% div_exp_eq
tff(fact_1536_div__exp__eq,axiom,
! [A: int,Mb: nat,Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_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,Mb),Nb))) ).
% div_exp_eq
tff(fact_1537_field__less__half__sum,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).
% field_less_half_sum
tff(fact_1538_field__less__half__sum,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).
% field_less_half_sum
tff(fact_1539_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).
% vebt_insert.simps(4)
tff(fact_1540_div__nat__eqI,axiom,
! [Nb: nat,Q3: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q3)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q3 ) ) ) ).
% div_nat_eqI
tff(fact_1541_div__by__1,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).
% div_by_1
tff(fact_1542_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_1543_div__by__1,axiom,
! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).
% div_by_1
tff(fact_1544_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_1545_div__by__1,axiom,
! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).
% div_by_1
tff(fact_1546_real__divide__square__eq,axiom,
! [R2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R2) ).
% real_divide_square_eq
tff(fact_1547_power__minus__is__div,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1548_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_1549_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_1550_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_1551_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_1552_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_1553_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).
% add_diff_cancel_right
tff(fact_1554_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_1555_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_1556_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_1557_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_1558_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_1559_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_1560_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_1561_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).
% add_diff_cancel_left
tff(fact_1562_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_1563_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_1564_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_1565_diff__add__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).
% diff_add_cancel
tff(fact_1566_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_1567_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_1568_add__diff__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).
% add_diff_cancel
tff(fact_1569_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_1570_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_1571_bits__div__by__1,axiom,
! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).
% bits_div_by_1
tff(fact_1572_Suc__diff__diff,axiom,
! [Mb: nat,Nb: 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,Mb)),Nb)),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,Mb),Nb)),K) ).
% Suc_diff_diff
tff(fact_1573_diff__Suc__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).
% diff_Suc_Suc
tff(fact_1574_diff__diff__cancel,axiom,
! [I: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).
% diff_diff_cancel
tff(fact_1575_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_1576_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_1577_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1578_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(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_1579_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(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_1580_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_1581_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1582_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(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_1583_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(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_1584_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).
% right_diff_distrib_numeral
tff(fact_1585_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_1586_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).
% right_diff_distrib_numeral
tff(fact_1587_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_1588_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).
% left_diff_distrib_numeral
tff(fact_1589_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_1590_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)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).
% left_diff_distrib_numeral
tff(fact_1591_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_1592_diff__Suc__1,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).
% diff_Suc_1
tff(fact_1593_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(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_1594_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(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_1595_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(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_1596_option_Ocollapse,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).
% option.collapse
tff(fact_1597_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( some_nat(the_nat(Option)) = Option ) ) ).
% option.collapse
tff(fact_1598_option_Ocollapse,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( some_num(the_num(Option)) = Option ) ) ).
% option.collapse
tff(fact_1599_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(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_1600_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(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_1601_less__eq__real__def,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
| ( Xb = Y ) ) ) ).
% less_eq_real_def
tff(fact_1602_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_1603_diff__right__commute,axiom,
! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).
% diff_right_commute
tff(fact_1604_diff__right__commute,axiom,
! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).
% diff_right_commute
tff(fact_1605_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_1606_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_1607_diff__eq__diff__eq,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) )
=> ( ( A = B )
<=> ( C = D ) ) ) ).
% diff_eq_diff_eq
tff(fact_1608_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
=> ( ( A = B )
<=> ( C = D ) ) ) ).
% diff_eq_diff_eq
tff(fact_1609_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_1610_real__arch__pow,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ).
% real_arch_pow
tff(fact_1611_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_1612_left__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).
% left_diff_distrib
tff(fact_1613_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_1614_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_1615_right__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).
% right_diff_distrib
tff(fact_1616_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_1617_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_1618_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).
% left_diff_distrib'
tff(fact_1619_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_1620_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_1621_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_1622_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).
% right_diff_distrib'
tff(fact_1623_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_1624_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_1625_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_1626_add__diff__add,axiom,
! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).
% add_diff_add
tff(fact_1627_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_1628_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1629_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) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1630_diff__eq__diff__less__eq,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) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1631_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).
% diff_right_mono
tff(fact_1632_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_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_1633_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).
% diff_right_mono
tff(fact_1634_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).
% diff_left_mono
tff(fact_1635_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> aa_int_o(aa_int_fun_int_o(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_1636_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).
% diff_left_mono
tff(fact_1637_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).
% diff_mono
tff(fact_1638_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
=> aa_int_o(aa_int_fun_int_o(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_1639_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).
% diff_mono
tff(fact_1640_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_1641_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).
% diff_strict_right_mono
tff(fact_1642_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,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_1643_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> aa_real_o(aa_real_fun_real_o(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_1644_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).
% diff_strict_left_mono
tff(fact_1645_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> aa_int_o(aa_int_fun_int_o(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_1646_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) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).
% diff_eq_diff_less
tff(fact_1647_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).
% diff_eq_diff_less
tff(fact_1648_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) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).
% diff_eq_diff_less
tff(fact_1649_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
=> aa_real_o(aa_real_fun_real_o(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_1650_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).
% diff_strict_mono
tff(fact_1651_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
=> aa_int_o(aa_int_fun_int_o(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_1652_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_1653_diff__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).
% diff_diff_eq
tff(fact_1654_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_1655_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_1656_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_1657_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( aa_rat_rat(plus_plus_rat(C),B) = A )
=> ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).
% add_implies_diff
tff(fact_1658_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_1659_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_1660_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_1661_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_1662_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_1663_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_1664_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).
% diff_add_eq
tff(fact_1665_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_1666_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_1667_diff__diff__eq2,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).
% diff_diff_eq2
tff(fact_1668_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_1669_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_1670_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).
% add_diff_eq
tff(fact_1671_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_1672_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_1673_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A = aa_rat_rat(minus_minus_rat(C),B) )
<=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).
% eq_diff_eq
tff(fact_1674_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_1675_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_1676_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(minus_minus_rat(A),B) = C )
<=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).
% diff_eq_eq
tff(fact_1677_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_1678_group__cancel_Osub1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A3),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_1679_group__cancel_Osub1,axiom,
! [A3: rat,K: rat,A: rat,B: rat] :
( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
=> ( aa_rat_rat(minus_minus_rat(A3),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).
% group_cancel.sub1
tff(fact_1680_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),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_1681_zero__induct__lemma,axiom,
! [P: fun_nat_o,K: nat,I: nat] :
( aa_nat_o(P,K)
=> ( ! [N: nat] :
( aa_nat_o(P,aa_nat_nat(suc,N))
=> aa_nat_o(P,N) )
=> aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).
% zero_induct_lemma
tff(fact_1682_less__imp__diff__less,axiom,
! [J: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K) ) ).
% less_imp_diff_less
tff(fact_1683_diff__less__mono2,axiom,
! [Mb: nat,Nb: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).
% diff_less_mono2
tff(fact_1684_diff__le__mono2,axiom,
! [Mb: nat,Nb: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).
% diff_le_mono2
tff(fact_1685_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).
% le_diff_iff'
tff(fact_1686_diff__le__self,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).
% diff_le_self
tff(fact_1687_diff__le__mono,axiom,
! [Mb: nat,Nb: nat,L: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).
% diff_le_mono
tff(fact_1688_Nat_Odiff__diff__eq,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).
% Nat.diff_diff_eq
tff(fact_1689_le__diff__iff,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).
% le_diff_iff
tff(fact_1690_eq__diff__iff,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
<=> ( Mb = Nb ) ) ) ) ).
% eq_diff_iff
tff(fact_1691_diff__add__inverse2,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).
% diff_add_inverse2
tff(fact_1692_diff__add__inverse,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).
% diff_add_inverse
tff(fact_1693_diff__cancel2,axiom,
! [Mb: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).
% diff_cancel2
tff(fact_1694_Nat_Odiff__cancel,axiom,
! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).
% Nat.diff_cancel
tff(fact_1695_diff__mult__distrib2,axiom,
! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).
% diff_mult_distrib2
tff(fact_1696_diff__mult__distrib,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).
% diff_mult_distrib
tff(fact_1697_add__le__add__imp__diff__le,axiom,
! [I: rat,K: rat,Nb: rat,J: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K)),J) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1698_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,Nb: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),J) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1699_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,Nb: int,J: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)),J) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1700_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,Nb: real,J: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)),J) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1701_add__le__imp__le__diff,axiom,
! [I: rat,K: rat,Nb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).
% add_le_imp_le_diff
tff(fact_1702_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).
% add_le_imp_le_diff
tff(fact_1703_add__le__imp__le__diff,axiom,
! [I: int,K: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)) ) ).
% add_le_imp_le_diff
tff(fact_1704_add__le__imp__le__diff,axiom,
! [I: real,K: real,Nb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)) ) ).
% add_le_imp_le_diff
tff(fact_1705_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ aa_real_o(aa_real_fun_real_o(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_1706_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_1707_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(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_1708_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(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_1709_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_1710_eq__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
<=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C) = D ) ) ).
% eq_add_iff1
tff(fact_1711_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_1712_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_1713_eq__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
<=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D) ) ) ).
% eq_add_iff2
tff(fact_1714_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_1715_square__diff__square__factored,axiom,
! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).
% square_diff_square_factored
tff(fact_1716_square__diff__square__factored,axiom,
! [Xb: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_rat_rat(minus_minus_rat(Xb),Y)) ).
% square_diff_square_factored
tff(fact_1717_square__diff__square__factored,axiom,
! [Xb: 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,Xb),Xb)),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,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).
% square_diff_square_factored
tff(fact_1718_mult__diff__mult,axiom,
! [Xb: 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,Xb),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,Xb),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,Xb),A)),B)) ).
% mult_diff_mult
tff(fact_1719_mult__diff__mult,axiom,
! [Xb: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),A)),B)) ).
% mult_diff_mult
tff(fact_1720_mult__diff__mult,axiom,
! [Xb: 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,Xb),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,Xb),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,Xb),A)),B)) ).
% mult_diff_mult
tff(fact_1721_option_Osel,axiom,
! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).
% option.sel
tff(fact_1722_option_Osel,axiom,
! [X2: nat] : the_nat(some_nat(X2)) = X2 ).
% option.sel
tff(fact_1723_option_Osel,axiom,
! [X2: num] : the_num(some_num(X2)) = X2 ).
% option.sel
tff(fact_1724_option_Oexpand,axiom,
! [Option: option_nat,Option2: option_nat] :
( ( ( Option = none_nat )
<=> ( Option2 = none_nat ) )
=> ( ( ( Option != none_nat )
=> ( ( Option2 != none_nat )
=> ( the_nat(Option) = the_nat(Option2) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
tff(fact_1725_option_Oexpand,axiom,
! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
( ( ( Option = none_P5556105721700978146at_nat )
<=> ( Option2 = none_P5556105721700978146at_nat ) )
=> ( ( ( Option != none_P5556105721700978146at_nat )
=> ( ( Option2 != none_P5556105721700978146at_nat )
=> ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
tff(fact_1726_option_Oexpand,axiom,
! [Option: option_num,Option2: option_num] :
( ( ( Option = none_num )
<=> ( Option2 = none_num ) )
=> ( ( ( Option != none_num )
=> ( ( Option2 != none_num )
=> ( the_num(Option) = the_num(Option2) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
tff(fact_1727_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D)) ) ).
% ordered_ring_class.le_add_iff2
tff(fact_1728_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(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_1729_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(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_o(aa_real_fun_real_o(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_1730_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D) ) ).
% ordered_ring_class.le_add_iff1
tff(fact_1731_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(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_1732_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(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_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(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_1733_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(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_1734_less__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D) ) ).
% less_add_iff1
tff(fact_1735_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(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_1736_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(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_1737_less__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D)) ) ).
% less_add_iff2
tff(fact_1738_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(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_1739_square__diff__one__factored,axiom,
! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Xb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xb),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ).
% square_diff_one_factored
tff(fact_1740_square__diff__one__factored,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ).
% square_diff_one_factored
tff(fact_1741_square__diff__one__factored,axiom,
! [Xb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ).
% square_diff_one_factored
tff(fact_1742_square__diff__one__factored,axiom,
! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),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,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).
% square_diff_one_factored
tff(fact_1743_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_1744_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1745_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1746_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1747_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1748_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1749_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1750_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(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_1751_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_1752_diff__add,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_1753_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).
% le_diff_eq
tff(fact_1754_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).
% le_diff_eq
tff(fact_1755_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).
% le_diff_eq
tff(fact_1756_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).
% diff_le_eq
tff(fact_1757_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).
% diff_le_eq
tff(fact_1758_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).
% diff_le_eq
tff(fact_1759_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).
% diff_less_eq
tff(fact_1760_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).
% diff_less_eq
tff(fact_1761_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).
% diff_less_eq
tff(fact_1762_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).
% less_diff_eq
tff(fact_1763_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).
% less_diff_eq
tff(fact_1764_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).
% less_diff_eq
tff(fact_1765_diff__less__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).
% diff_less_Suc
tff(fact_1766_Suc__diff__Suc,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).
% Suc_diff_Suc
tff(fact_1767_Suc__diff__le,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).
% Suc_diff_le
tff(fact_1768_diff__Suc__eq__diff__pred,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).
% diff_Suc_eq_diff_pred
tff(fact_1769_less__diff__iff,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).
% less_diff_iff
tff(fact_1770_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(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_1771_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).
% less_diff_conv
tff(fact_1772_add__diff__inverse__nat,axiom,
! [Mb: nat,Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).
% add_diff_inverse_nat
tff(fact_1773_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( ( aa_nat_nat(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_1774_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(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_1775_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_nat(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_1776_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).
% Nat.le_diff_conv2
tff(fact_1777_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).
% le_diff_conv
tff(fact_1778_option_Oexhaust__sel,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).
% option.exhaust_sel
tff(fact_1779_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option = some_nat(the_nat(Option)) ) ) ).
% option.exhaust_sel
tff(fact_1780_option_Oexhaust__sel,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( Option = some_num(the_num(Option)) ) ) ).
% option.exhaust_sel
tff(fact_1781_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).
% less_diff_conv2
tff(fact_1782_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),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)),Nb)) ) ) ).
% nat_diff_add_eq2
tff(fact_1783_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = 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)),Mb)),Nb) ) ) ).
% nat_diff_add_eq1
tff(fact_1784_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),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)),Nb)) ) ) ).
% nat_le_add_iff2
tff(fact_1785_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).
% nat_le_add_iff1
tff(fact_1786_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
<=> ( Mb = 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)),Nb) ) ) ) ).
% nat_eq_add_iff2
tff(fact_1787_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
<=> ( 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)),Mb) = Nb ) ) ) ).
% nat_eq_add_iff1
tff(fact_1788_linorder__neqE__linordered__idom,axiom,
! [Xb: real,Y: real] :
( ( Xb != Y )
=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_1789_linorder__neqE__linordered__idom,axiom,
! [Xb: rat,Y: rat] :
( ( Xb != Y )
=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_1790_linorder__neqE__linordered__idom,axiom,
! [Xb: int,Y: int] :
( ( Xb != Y )
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_1791_power2__commute,axiom,
! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),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),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1792_power2__commute,axiom,
! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),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),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1793_power2__commute,axiom,
! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1794_power2__commute,axiom,
! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),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),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1795_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),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)),Nb)) ) ) ).
% nat_less_add_iff2
tff(fact_1796_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).
% nat_less_add_iff1
tff(fact_1797_diff__le__diff__pow,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).
% diff_le_diff_pow
tff(fact_1798_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_1799_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E)),C) ).
% combine_common_factor
tff(fact_1800_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_1801_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_1802_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_1803_distrib__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).
% distrib_right
tff(fact_1804_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_1805_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_1806_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_1807_distrib__left,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).
% distrib_left
tff(fact_1808_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_1809_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_1810_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_1811_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(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_1812_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_1813_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_1814_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_1815_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(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_1816_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_1817_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_1818_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(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_1819_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_1820_two__realpow__ge__one,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).
% two_realpow_ge_one
tff(fact_1821_power2__diff,axiom,
! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)) ).
% power2_diff
tff(fact_1822_power2__diff,axiom,
! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),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(Xb),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))),Xb)),Y)) ).
% power2_diff
tff(fact_1823_power2__diff,axiom,
! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),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,Xb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)) ).
% power2_diff
tff(fact_1824_power2__diff,axiom,
! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),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(Xb),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))),Xb)),Y)) ).
% power2_diff
tff(fact_1825_div__le__dividend,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).
% div_le_dividend
tff(fact_1826_div__le__mono,axiom,
! [Mb: nat,Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).
% div_le_mono
tff(fact_1827_div__mult2__eq,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q3) ).
% div_mult2_eq
tff(fact_1828_less__1__mult,axiom,
! [Mb: real,Nb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),Nb)) ) ) ).
% less_1_mult
tff(fact_1829_less__1__mult,axiom,
! [Mb: rat,Nb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Mb),Nb)) ) ) ).
% less_1_mult
tff(fact_1830_less__1__mult,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).
% less_1_mult
tff(fact_1831_less__1__mult,axiom,
! [Mb: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) ) ) ).
% less_1_mult
tff(fact_1832_add__mono1,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_1833_add__mono1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).
% add_mono1
tff(fact_1834_add__mono1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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_1835_add__mono1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,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_1836_less__add__one,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).
% less_add_one
tff(fact_1837_less__add__one,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).
% less_add_one
tff(fact_1838_less__add__one,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).
% less_add_one
tff(fact_1839_less__add__one,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).
% less_add_one
tff(fact_1840_Suc__div__le__mono,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).
% Suc_div_le_mono
tff(fact_1841_less__mult__imp__div__less,axiom,
! [Mb: nat,I: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),I) ) ).
% less_mult_imp_div_less
tff(fact_1842_div__times__less__eq__dividend,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).
% div_times_less_eq_dividend
tff(fact_1843_times__div__less__eq__dividend,axiom,
! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).
% times_div_less_eq_dividend
tff(fact_1844_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_1845_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_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_1846_numeral__Bit0__div__2,axiom,
! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).
% numeral_Bit0_div_2
tff(fact_1847_numeral__Bit0__div__2,axiom,
! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).
% numeral_Bit0_div_2
tff(fact_1848_field__sum__of__halves,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xb ).
% field_sum_of_halves
tff(fact_1849_field__sum__of__halves,axiom,
! [Xb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xb ).
% field_sum_of_halves
tff(fact_1850_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_1851_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_1852_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).
% times_divide_eq_left
tff(fact_1853_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_1854_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_1855_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).
% divide_divide_eq_left
tff(fact_1856_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_1857_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_1858_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).
% divide_divide_eq_right
tff(fact_1859_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_1860_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_1861_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).
% times_divide_eq_right
tff(fact_1862_divmod__step__eq,axiom,
! [L: num,Q3: nat,R2: nat] :
unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),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))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))),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))),Q3)),R2)) ).
% divmod_step_eq
tff(fact_1863_divmod__step__eq,axiom,
! [L: num,Q3: int,R2: int] :
aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),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))),Q3)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),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))),Q3)),R2)) ).
% divmod_step_eq
tff(fact_1864_divmod__step__eq,axiom,
! [L: num,Q3: code_integer,R2: code_integer] :
unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),R2)) ).
% divmod_step_eq
tff(fact_1865_pred__less__length__list,axiom,
! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summarya,h),
$ite(
pr = none_nat,
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
tff(fact_1866_pred__lesseq__max,axiom,
! [Dega: nat,Xb: nat,Maa: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summarya,h),
$ite(
pr = none_nat,
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
none_nat ) ) ) ) ) ) ).
% pred_lesseq_max
tff(fact_1867_succ__greatereq__min,axiom,
! [Dega: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summarya,h),
$ite(sc = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
none_nat ) ) ) ) ) ) ).
% succ_greatereq_min
tff(fact_1868_succ__less__length__list,axiom,
! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summarya,h),
$ite(sc = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
tff(fact_1869_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_1870_set__vebt_H__def,axiom,
! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).
% set_vebt'_def
tff(fact_1871_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] : aa_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(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).
% zdiv_numeral_Bit0
tff(fact_1872_lambda__one,axiom,
aTP_Lamm_ag = times_times_complex(one_one_complex) ).
% lambda_one
tff(fact_1873_lambda__one,axiom,
aTP_Lamm_ah = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).
% lambda_one
tff(fact_1874_lambda__one,axiom,
aTP_Lamm_ai = times_times_rat(one_one_rat) ).
% lambda_one
tff(fact_1875_lambda__one,axiom,
aTP_Lamm_aj = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).
% lambda_one
tff(fact_1876_lambda__one,axiom,
aTP_Lamm_ak = aa_int_fun_int_int(times_times_int,one_one_int) ).
% lambda_one
tff(fact_1877_set__vebt__def,axiom,
! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).
% set_vebt_def
tff(fact_1878_numeral__code_I2_J,axiom,
! [Nb: num] :
aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
m2: rat,
m2:= aa_num_rat(numeral_numeral_rat,Nb),
aa_rat_rat(plus_plus_rat(m2),m2) ) ).
% numeral_code(2)
tff(fact_1879_numeral__code_I2_J,axiom,
! [Nb: num] :
numera1916890842035813515d_enat(aa_num_num(bit0,Nb)) = $let(
m2: extended_enat,
m2:= numera1916890842035813515d_enat(Nb),
plus_p3455044024723400733d_enat(m2,m2) ) ).
% numeral_code(2)
tff(fact_1880_numeral__code_I2_J,axiom,
! [Nb: num] :
numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
m2: complex,
m2:= numera6690914467698888265omplex(Nb),
aa_complex_complex(plus_plus_complex(m2),m2) ) ).
% numeral_code(2)
tff(fact_1881_numeral__code_I2_J,axiom,
! [Nb: num] :
aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
m2: real,
m2:= aa_num_real(numeral_numeral_real,Nb),
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).
% numeral_code(2)
tff(fact_1882_numeral__code_I2_J,axiom,
! [Nb: num] :
aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
m2: nat,
m2:= aa_num_nat(numeral_numeral_nat,Nb),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).
% numeral_code(2)
tff(fact_1883_numeral__code_I2_J,axiom,
! [Nb: num] :
aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
m2: int,
m2:= aa_num_int(numeral_numeral_int,Nb),
aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).
% numeral_code(2)
tff(fact_1884_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
=> ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Z) ) ) ).
% add_diff_assoc_enat
tff(fact_1885_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))) = $let(
w: complex,
w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
aa_complex_complex(times_times_complex(w),w) ) ).
% power_numeral_even
tff(fact_1886_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))) = $let(
w: real,
w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).
% power_numeral_even
tff(fact_1887_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))) = $let(
w: rat,
w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
aa_rat_rat(times_times_rat(w),w) ) ).
% power_numeral_even
tff(fact_1888_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))) = $let(
w: nat,
w:= 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(times_times_nat,w),w) ) ).
% power_numeral_even
tff(fact_1889_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))) = $let(
w: int,
w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).
% power_numeral_even
tff(fact_1890_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_13) ).
% linordered_field_no_ub
tff(fact_1891_linordered__field__no__ub,axiom,
! [X3: rat] :
? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_13) ).
% linordered_field_no_ub
tff(fact_1892_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X3) ).
% linordered_field_no_lb
tff(fact_1893_linordered__field__no__lb,axiom,
! [X3: rat] :
? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X3) ).
% linordered_field_no_lb
tff(fact_1894_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_1895_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_1896_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).
% divide_divide_eq_left'
tff(fact_1897_divide__divide__times__eq,axiom,
! [Xb: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),W)),aa_complex_complex(times_times_complex(Y),Z)) ).
% divide_divide_times_eq
tff(fact_1898_divide__divide__times__eq,axiom,
! [Xb: 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,Xb),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,Xb),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).
% divide_divide_times_eq
tff(fact_1899_divide__divide__times__eq,axiom,
! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),W)),aa_rat_rat(times_times_rat(Y),Z)) ).
% divide_divide_times_eq
tff(fact_1900_times__divide__times__eq,axiom,
! [Xb: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(Y),W)) ).
% times_divide_times_eq
tff(fact_1901_times__divide__times__eq,axiom,
! [Xb: 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,Xb),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,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).
% times_divide_times_eq
tff(fact_1902_times__divide__times__eq,axiom,
! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),W)) ).
% times_divide_times_eq
tff(fact_1903_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_1904_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_1905_add__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).
% add_divide_distrib
tff(fact_1906_vebt__pred_Osimps_I7_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
some_nat(Maa),
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
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(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summarya,h),
$ite(
pr = none_nat,
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),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)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
none_nat ) ) ) ) ).
% vebt_pred.simps(7)
tff(fact_1907_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeLista,h)),l)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summarya),h)),one_one_nat) ) ),
one_one_nat ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_1908_is__pred__in__set__def,axiom,
! [Xs: set_nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xb),Y)
<=> ( aa_set_nat_o(member_nat(Y),Xs)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
& ! [X: nat] :
( aa_set_nat_o(member_nat(X),Xs)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ).
% is_pred_in_set_def
tff(fact_1909_discrete,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).
% discrete
tff(fact_1910_discrete,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).
% discrete
tff(fact_1911_gt__half__sum,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_1912_gt__half__sum,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).
% gt_half_sum
tff(fact_1913_less__half__sum,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(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_1914_less__half__sum,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).
% less_half_sum
tff(fact_1915_vebt__succ_Osimps_I6_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
some_nat(Mia),
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
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(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summarya,h),
$ite(sc = 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)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
none_nat ) ) ) ) ).
% vebt_succ.simps(6)
tff(fact_1916_vebt__member_Osimps_I5_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb)
<=> $ite(
Xb = Mia,
$true,
$ite(
Xb = Maa,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).
% vebt_member.simps(5)
tff(fact_1917_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)),aa_num_real(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% real_average_minus_second
tff(fact_1918_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)),aa_num_real(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% real_average_minus_first
tff(fact_1919_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeLista,h)),l)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summarya),h)),one_one_nat) ) ),
one_one_nat ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_1920_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,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
Xb = Mia,
zero_zero_nat,
$ite(
Xb = Maa,
zero_zero_nat,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
zero_zero_nat,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
zero_zero_nat,
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) ),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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_1921_pred__empty,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_pred(Ta,Xb) = none_nat )
<=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_al(Ta),Xb)) = bot_bot_set_nat ) ) ) ).
% pred_empty
tff(fact_1922_succ__empty,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_succ(Ta,Xb) = none_nat )
<=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_am(Ta),Xb)) = bot_bot_set_nat ) ) ) ).
% succ_empty
tff(fact_1923_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
<=> ( ( Xb = Mia )
| ( Xb = Maa )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).
% VEBT_internal.membermima.simps(4)
tff(fact_1924_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S2)),Xb)
<=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).
% VEBT_internal.naive_member.simps(3)
tff(fact_1925_valid__0__not,axiom,
! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).
% valid_0_not
tff(fact_1926_valid__tree__deg__neq__0,axiom,
! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).
% valid_tree_deg_neq_0
tff(fact_1927_deg__not__0,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% deg_not_0
tff(fact_1928_both__member__options__def,axiom,
! [Ta: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
<=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xb)
| aa_nat_o(vEBT_VEBT_membermima(Ta),Xb) ) ) ).
% both_member_options_def
tff(fact_1929_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
=> ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
=> ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
| aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).
% member_valid_both_member_options
tff(fact_1930_mint__corr__help__empty,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_mint(Ta) = none_nat )
=> ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).
% mint_corr_help_empty
tff(fact_1931_maxt__corr__help__empty,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( vEBT_vebt_maxt(Ta) = none_nat )
=> ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).
% maxt_corr_help_empty
tff(fact_1932_le__zero__eq,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
<=> ( Nb = zero_zero_nat ) ) ).
% le_zero_eq
tff(fact_1933_not__gr__zero,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
<=> ( Nb = zero_zero_nat ) ) ).
% not_gr_zero
tff(fact_1934_mult__zero__left,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).
% mult_zero_left
tff(fact_1935_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_1936_mult__zero__left,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).
% mult_zero_left
tff(fact_1937_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_1938_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_1939_mult__zero__right,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).
% mult_zero_right
tff(fact_1940_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_1941_mult__zero__right,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).
% mult_zero_right
tff(fact_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_add__0,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).
% add_0
tff(fact_1960_add__0,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).
% add_0
tff(fact_1961_add__0,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).
% add_0
tff(fact_1962_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_1963_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_1964_zero__eq__add__iff__both__eq__0,axiom,
! [Xb: nat,Y: nat] :
( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
<=> ( ( Xb = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_1965_add__eq__0__iff__both__eq__0,axiom,
! [Xb: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
<=> ( ( Xb = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_1966_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_1967_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_1968_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(plus_plus_rat(A),B) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
tff(fact_1969_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_1970_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_1971_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_1972_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_1973_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(plus_plus_rat(B),A) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
tff(fact_1974_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_1975_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_1976_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_1977_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_1978_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(plus_plus_rat(A),B) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
tff(fact_1979_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_1980_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_1981_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_1982_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_1983_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( aa_rat_rat(plus_plus_rat(B),A) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
tff(fact_1984_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_1985_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_1986_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_1987_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
<=> ( A = zero_zero_rat ) ) ).
% double_zero_sym
tff(fact_1988_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_1989_add_Oright__neutral,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).
% add.right_neutral
tff(fact_1990_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_1991_add_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).
% add.right_neutral
tff(fact_1992_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_1993_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_1994_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_1995_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_1996_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1997_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_1998_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_1999_diff__zero,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).
% diff_zero
tff(fact_2000_diff__zero,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).
% diff_zero
tff(fact_2001_diff__zero,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).
% diff_zero
tff(fact_2002_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_2003_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_2004_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_2005_diff__0__right,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).
% diff_0_right
tff(fact_2006_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_2007_diff__0__right,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).
% diff_0_right
tff(fact_2008_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_2009_diff__self,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).
% diff_self
tff(fact_2010_diff__self,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).
% diff_self
tff(fact_2011_diff__self,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).
% diff_self
tff(fact_2012_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_2013_less__nat__zero__code,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).
% less_nat_zero_code
tff(fact_2014_neq0__conv,axiom,
! [Nb: nat] :
( ( Nb != zero_zero_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% neq0_conv
tff(fact_2015_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).
% bot_nat_0.not_eq_extremum
tff(fact_2016_le0,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).
% le0
tff(fact_2017_bot__nat__0_Oextremum,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).
% bot_nat_0.extremum
tff(fact_2018_Nat_Oadd__0__right,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).
% Nat.add_0_right
tff(fact_2019_add__is__0,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
<=> ( ( Mb = zero_zero_nat )
& ( Nb = zero_zero_nat ) ) ) ).
% add_is_0
tff(fact_2020_diff__self__eq__0,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).
% diff_self_eq_0
tff(fact_2021_diff__0__eq__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).
% diff_0_eq_0
tff(fact_2022_mult__cancel2,axiom,
! [Mb: nat,K: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
<=> ( ( Mb = Nb )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
tff(fact_2023_mult__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
<=> ( ( Mb = Nb )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
tff(fact_2024_mult__0__right,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).
% mult_0_right
tff(fact_2025_mult__is__0,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
<=> ( ( Mb = zero_zero_nat )
| ( Nb = zero_zero_nat ) ) ) ).
% mult_is_0
tff(fact_2026_dbl__simps_I2_J,axiom,
aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).
% dbl_simps(2)
tff(fact_2027_dbl__simps_I2_J,axiom,
aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).
% dbl_simps(2)
tff(fact_2028_dbl__simps_I2_J,axiom,
aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).
% dbl_simps(2)
tff(fact_2029_dbl__simps_I2_J,axiom,
aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).
% dbl_simps(2)
tff(fact_2030_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_2031_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_2032_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_2033_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_2034_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_2035_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_2036_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).
% le_add_same_cancel2
tff(fact_2037_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).
% le_add_same_cancel2
tff(fact_2038_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).
% le_add_same_cancel2
tff(fact_2039_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).
% le_add_same_cancel2
tff(fact_2040_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).
% le_add_same_cancel1
tff(fact_2041_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).
% le_add_same_cancel1
tff(fact_2042_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).
% le_add_same_cancel1
tff(fact_2043_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).
% le_add_same_cancel1
tff(fact_2044_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% add_le_same_cancel2
tff(fact_2045_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).
% add_le_same_cancel2
tff(fact_2046_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% add_le_same_cancel2
tff(fact_2047_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% add_le_same_cancel2
tff(fact_2048_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% add_le_same_cancel1
tff(fact_2049_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).
% add_le_same_cancel1
tff(fact_2050_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% add_le_same_cancel1
tff(fact_2051_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% add_le_same_cancel1
tff(fact_2052_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% add_less_same_cancel1
tff(fact_2053_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% add_less_same_cancel1
tff(fact_2054_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).
% add_less_same_cancel1
tff(fact_2055_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% add_less_same_cancel1
tff(fact_2056_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% add_less_same_cancel2
tff(fact_2057_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% add_less_same_cancel2
tff(fact_2058_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).
% add_less_same_cancel2
tff(fact_2059_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% add_less_same_cancel2
tff(fact_2060_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).
% less_add_same_cancel1
tff(fact_2061_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).
% less_add_same_cancel1
tff(fact_2062_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).
% less_add_same_cancel1
tff(fact_2063_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).
% less_add_same_cancel1
tff(fact_2064_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).
% less_add_same_cancel2
tff(fact_2065_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).
% less_add_same_cancel2
tff(fact_2066_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).
% less_add_same_cancel2
tff(fact_2067_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).
% less_add_same_cancel2
tff(fact_2068_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_2069_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_2070_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_2071_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_2072_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_2073_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_2074_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).
% diff_ge_0_iff_ge
tff(fact_2075_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% diff_ge_0_iff_ge
tff(fact_2076_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).
% diff_ge_0_iff_ge
tff(fact_2077_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).
% diff_gt_0_iff_gt
tff(fact_2078_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).
% diff_gt_0_iff_gt
tff(fact_2079_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).
% diff_gt_0_iff_gt
tff(fact_2080_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_2081_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_2082_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_2083_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_2084_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_2085_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_2086_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_2087_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_2088_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_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_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_2096_sum__squares__eq__zero__iff,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
<=> ( ( Xb = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_2097_sum__squares__eq__zero__iff,axiom,
! [Xb: rat,Y: rat] :
( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
<=> ( ( Xb = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_2098_sum__squares__eq__zero__iff,axiom,
! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
<=> ( ( Xb = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_2099_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_2100_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_2101_diff__numeral__special_I9_J,axiom,
aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).
% diff_numeral_special(9)
tff(fact_2102_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_2103_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_2104_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: 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)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).
% div_mult_mult1_if
tff(fact_2105_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
aa_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)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(divide_divide_int(A),B)) ).
% div_mult_mult1_if
tff(fact_2106_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_2107_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B) ) ) ).
% div_mult_mult2
tff(fact_2108_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_2109_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B) ) ) ).
% div_mult_mult1
tff(fact_2110_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_2111_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_2112_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2113_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_2114_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_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_2115_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_2116_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_2117_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2118_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_2119_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_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_2120_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_2121_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_2122_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_2123_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_2124_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_2125_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2126_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_2127_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_2128_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2129_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_2130_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_2131_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2132_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).
% mult_divide_mult_cancel_left_if
tff(fact_2133_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: 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)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).
% mult_divide_mult_cancel_left_if
tff(fact_2134_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B)) ).
% mult_divide_mult_cancel_left_if
tff(fact_2135_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).
% div_self
tff(fact_2136_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_2137_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).
% div_self
tff(fact_2138_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_2139_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).
% div_self
tff(fact_2140_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_2141_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
<=> ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
tff(fact_2142_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_2143_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
tff(fact_2144_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_2145_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
<=> ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
tff(fact_2146_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_2147_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
<=> ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
tff(fact_2148_divide__self__if,axiom,
! [A: complex] :
aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).
% divide_self_if
tff(fact_2149_divide__self__if,axiom,
! [A: real] :
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).
% divide_self_if
tff(fact_2150_divide__self__if,axiom,
! [A: rat] :
aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).
% divide_self_if
tff(fact_2151_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).
% divide_self
tff(fact_2152_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_2153_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).
% divide_self
tff(fact_2154_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_2155_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_2156_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
<=> ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
tff(fact_2157_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_2158_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_2159_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
<=> ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
tff(fact_2160_power__0__Suc,axiom,
! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).
% power_0_Suc
tff(fact_2161_power__0__Suc,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).
% power_0_Suc
tff(fact_2162_power__0__Suc,axiom,
! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).
% power_0_Suc
tff(fact_2163_power__0__Suc,axiom,
! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).
% power_0_Suc
tff(fact_2164_power__0__Suc,axiom,
! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).
% power_0_Suc
tff(fact_2165_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_2166_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_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_zero__less__Suc,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).
% zero_less_Suc
tff(fact_2175_less__Suc0,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
<=> ( Nb = zero_zero_nat ) ) ).
% less_Suc0
tff(fact_2176_add__gr__0,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% add_gr_0
tff(fact_2177_less__one,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
<=> ( Nb = zero_zero_nat ) ) ).
% less_one
tff(fact_2178_one__eq__mult__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
<=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
& ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% one_eq_mult_iff
tff(fact_2179_mult__eq__1__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
& ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% mult_eq_1_iff
tff(fact_2180_div__by__Suc__0,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).
% div_by_Suc_0
tff(fact_2181_zero__less__diff,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% zero_less_diff
tff(fact_2182_nat__mult__less__cancel__disj,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% nat_mult_less_cancel_disj
tff(fact_2183_nat__0__less__mult__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% nat_0_less_mult_iff
tff(fact_2184_mult__less__cancel2,axiom,
! [Mb: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% mult_less_cancel2
tff(fact_2185_div__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).
% div_less
tff(fact_2186_diff__is__0__eq,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% diff_is_0_eq
tff(fact_2187_diff__is__0__eq_H,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).
% diff_is_0_eq'
tff(fact_2188_nat__power__eq__Suc__0__iff,axiom,
! [Xb: nat,Mb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( Mb = zero_zero_nat )
| ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% nat_power_eq_Suc_0_iff
tff(fact_2189_power__Suc__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).
% power_Suc_0
tff(fact_2190_nat__zero__less__power__iff,axiom,
! [Xb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
| ( Nb = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
tff(fact_2191_nat__mult__div__cancel__disj,axiom,
! [K: nat,Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).
% nat_mult_div_cancel_disj
tff(fact_2192_zero__le__divide__1__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% zero_le_divide_1_iff
tff(fact_2193_zero__le__divide__1__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% zero_le_divide_1_iff
tff(fact_2194_divide__le__0__1__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% divide_le_0_1_iff
tff(fact_2195_divide__le__0__1__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% divide_le_0_1_iff
tff(fact_2196_zero__less__divide__1__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).
% zero_less_divide_1_iff
tff(fact_2197_zero__less__divide__1__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).
% zero_less_divide_1_iff
tff(fact_2198_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% less_divide_eq_1_pos
tff(fact_2199_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% less_divide_eq_1_pos
tff(fact_2200_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).
% less_divide_eq_1_neg
tff(fact_2201_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).
% less_divide_eq_1_neg
tff(fact_2202_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).
% divide_less_eq_1_pos
tff(fact_2203_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).
% divide_less_eq_1_pos
tff(fact_2204_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% divide_less_eq_1_neg
tff(fact_2205_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% divide_less_eq_1_neg
tff(fact_2206_divide__less__0__1__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% divide_less_0_1_iff
tff(fact_2207_divide__less__0__1__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% divide_less_0_1_iff
tff(fact_2208_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
<=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2209_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) )
<=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2210_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
<=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2211_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W: num,A: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
<=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2212_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) = A )
<=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2213_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
<=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2214_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_2215_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_2216_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2217_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_2218_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_2219_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2220_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_2221_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B)) ) ) ).
% div_mult_self1
tff(fact_2222_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_2223_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B)) ) ) ).
% div_mult_self2
tff(fact_2224_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_2225_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B)) ) ) ).
% div_mult_self3
tff(fact_2226_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_2227_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B)) ) ) ).
% div_mult_self4
tff(fact_2228_power__eq__0__iff,axiom,
! [A: rat,Nb: nat] :
( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% power_eq_0_iff
tff(fact_2229_power__eq__0__iff,axiom,
! [A: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% power_eq_0_iff
tff(fact_2230_power__eq__0__iff,axiom,
! [A: real,Nb: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
<=> ( ( A = zero_zero_real )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% power_eq_0_iff
tff(fact_2231_power__eq__0__iff,axiom,
! [A: int,Nb: nat] :
( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
<=> ( ( A = zero_zero_int )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% power_eq_0_iff
tff(fact_2232_power__eq__0__iff,axiom,
! [A: complex,Nb: nat] :
( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
<=> ( ( A = zero_zero_complex )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% power_eq_0_iff
tff(fact_2233_Suc__pred,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).
% Suc_pred
tff(fact_2234_one__le__mult__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).
% one_le_mult_iff
tff(fact_2235_mult__le__cancel2,axiom,
! [Mb: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% mult_le_cancel2
tff(fact_2236_nat__mult__le__cancel__disj,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% nat_mult_le_cancel_disj
tff(fact_2237_div__mult__self__is__m,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).
% div_mult_self_is_m
tff(fact_2238_div__mult__self1__is__m,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).
% div_mult_self1_is_m
tff(fact_2239_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% le_divide_eq_1_pos
tff(fact_2240_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% le_divide_eq_1_pos
tff(fact_2241_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).
% le_divide_eq_1_neg
tff(fact_2242_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).
% le_divide_eq_1_neg
tff(fact_2243_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).
% divide_le_eq_1_pos
tff(fact_2244_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).
% divide_le_eq_1_pos
tff(fact_2245_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% divide_le_eq_1_neg
tff(fact_2246_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% divide_le_eq_1_neg
tff(fact_2247_power__strict__decreasing__iff,axiom,
! [B: real,Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2248_power__strict__decreasing__iff,axiom,
! [B: rat,Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2249_power__strict__decreasing__iff,axiom,
! [B: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2250_power__strict__decreasing__iff,axiom,
! [B: int,Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2251_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_2252_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_2253_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_2254_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_2255_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_2256_power__mono__iff,axiom,
! [A: rat,B: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).
% power_mono_iff
tff(fact_2257_power__mono__iff,axiom,
! [A: nat,B: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).
% power_mono_iff
tff(fact_2258_power__mono__iff,axiom,
! [A: int,B: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).
% power_mono_iff
tff(fact_2259_power__mono__iff,axiom,
! [A: real,B: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).
% power_mono_iff
tff(fact_2260_Suc__diff__1,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).
% Suc_diff_1
tff(fact_2261_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_2262_bits__1__div__2,axiom,
aa_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_2263_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_2264_one__div__two__eq__zero,axiom,
aa_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_2265_power__decreasing__iff,axiom,
! [B: rat,Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).
% power_decreasing_iff
tff(fact_2266_power__decreasing__iff,axiom,
! [B: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).
% power_decreasing_iff
tff(fact_2267_power__decreasing__iff,axiom,
! [B: int,Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).
% power_decreasing_iff
tff(fact_2268_power__decreasing__iff,axiom,
! [B: real,Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).
% power_decreasing_iff
tff(fact_2269_power2__less__eq__zero__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(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_2270_power2__less__eq__zero__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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_2271_power2__less__eq__zero__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(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_2272_power2__eq__iff__nonneg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( ( aa_nat_rat(power_power_rat(Xb),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))) )
<=> ( Xb = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2273_power2__eq__iff__nonneg,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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))) )
<=> ( Xb = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2274_power2__eq__iff__nonneg,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( ( aa_nat_int(power_power_int(Xb),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))) )
<=> ( Xb = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2275_power2__eq__iff__nonneg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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))) )
<=> ( Xb = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2276_zero__less__power2,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(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_2277_zero__less__power2,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(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_2278_zero__less__power2,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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_2279_sum__power2__eq__zero__iff,axiom,
! [Xb: rat,Y: rat] :
( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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 )
<=> ( ( Xb = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2280_sum__power2__eq__zero__iff,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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 )
<=> ( ( Xb = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2281_sum__power2__eq__zero__iff,axiom,
! [Xb: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),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 )
<=> ( ( Xb = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2282_zero__reorient,axiom,
! [Xb: complex] :
( ( zero_zero_complex = Xb )
<=> ( Xb = zero_zero_complex ) ) ).
% zero_reorient
tff(fact_2283_zero__reorient,axiom,
! [Xb: real] :
( ( zero_zero_real = Xb )
<=> ( Xb = zero_zero_real ) ) ).
% zero_reorient
tff(fact_2284_zero__reorient,axiom,
! [Xb: rat] :
( ( zero_zero_rat = Xb )
<=> ( Xb = zero_zero_rat ) ) ).
% zero_reorient
tff(fact_2285_zero__reorient,axiom,
! [Xb: nat] :
( ( zero_zero_nat = Xb )
<=> ( Xb = zero_zero_nat ) ) ).
% zero_reorient
tff(fact_2286_zero__reorient,axiom,
! [Xb: int] :
( ( zero_zero_int = Xb )
<=> ( Xb = zero_zero_int ) ) ).
% zero_reorient
tff(fact_2287_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Ux) ).
% VEBT_internal.naive_member.simps(2)
tff(fact_2288_power__0__left,axiom,
! [Nb: nat] :
aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).
% power_0_left
tff(fact_2289_power__0__left,axiom,
! [Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).
% power_0_left
tff(fact_2290_power__0__left,axiom,
! [Nb: nat] :
aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).
% power_0_left
tff(fact_2291_power__0__left,axiom,
! [Nb: nat] :
aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).
% power_0_left
tff(fact_2292_power__0__left,axiom,
! [Nb: nat] :
aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).
% power_0_left
tff(fact_2293_bot_Oextremum__uniqueI,axiom,
! [A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),bot_bo4199563552545308370d_enat)
=> ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
tff(fact_2294_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
tff(fact_2295_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
tff(fact_2296_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
tff(fact_2297_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
tff(fact_2298_bot_Oextremum__unique,axiom,
! [A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),bot_bo4199563552545308370d_enat)
<=> ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
tff(fact_2299_bot_Oextremum__unique,axiom,
! [A: set_int] :
( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
<=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
tff(fact_2300_bot_Oextremum__unique,axiom,
! [A: set_real] :
( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
<=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
tff(fact_2301_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
<=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
tff(fact_2302_bot_Oextremum__unique,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
<=> ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
tff(fact_2303_bot_Oextremum,axiom,
! [A: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(bot_bo4199563552545308370d_enat),A) ).
% bot.extremum
tff(fact_2304_bot_Oextremum,axiom,
! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).
% bot.extremum
tff(fact_2305_bot_Oextremum,axiom,
! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).
% bot.extremum
tff(fact_2306_bot_Oextremum,axiom,
! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).
% bot.extremum
tff(fact_2307_bot_Oextremum,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).
% bot.extremum
tff(fact_2308_zero__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).
% zero_power
tff(fact_2309_zero__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).
% zero_power
tff(fact_2310_zero__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).
% zero_power
tff(fact_2311_zero__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).
% zero_power
tff(fact_2312_zero__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).
% zero_power
tff(fact_2313_bot_Oextremum__strict,axiom,
! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).
% bot.extremum_strict
tff(fact_2314_bot_Oextremum__strict,axiom,
! [A: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(A),bot_bo4199563552545308370d_enat) ).
% bot.extremum_strict
tff(fact_2315_bot_Oextremum__strict,axiom,
! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).
% bot.extremum_strict
tff(fact_2316_bot_Oextremum__strict,axiom,
! [A: set_real] : ~ aa_set_real_o(ord_less_set_real(A),bot_bot_set_real) ).
% bot.extremum_strict
tff(fact_2317_bot_Oextremum__strict,axiom,
! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).
% bot.extremum_strict
tff(fact_2318_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
<=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A) ) ).
% bot.not_eq_extremum
tff(fact_2319_bot_Onot__eq__extremum,axiom,
! [A: extended_enat] :
( ( A != bot_bo4199563552545308370d_enat )
<=> aa_Extended_enat_o(ord_le72135733267957522d_enat(bot_bo4199563552545308370d_enat),A) ) ).
% bot.not_eq_extremum
tff(fact_2320_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
<=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).
% bot.not_eq_extremum
tff(fact_2321_bot_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
<=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),A) ) ).
% bot.not_eq_extremum
tff(fact_2322_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).
% bot.not_eq_extremum
tff(fact_2323_zero__le,axiom,
! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).
% zero_le
tff(fact_2324_le__numeral__extra_I3_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).
% le_numeral_extra(3)
tff(fact_2325_le__numeral__extra_I3_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).
% le_numeral_extra(3)
tff(fact_2326_le__numeral__extra_I3_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).
% le_numeral_extra(3)
tff(fact_2327_le__numeral__extra_I3_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).
% le_numeral_extra(3)
tff(fact_2328_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
=> ? [E2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
& aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D2) ) ) ) ).
% field_lbound_gt_zero
tff(fact_2329_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D2)
=> ? [E2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D1)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D2) ) ) ) ).
% field_lbound_gt_zero
tff(fact_2330_less__numeral__extra_I3_J,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).
% less_numeral_extra(3)
tff(fact_2331_less__numeral__extra_I3_J,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).
% less_numeral_extra(3)
tff(fact_2332_less__numeral__extra_I3_J,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).
% less_numeral_extra(3)
tff(fact_2333_less__numeral__extra_I3_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).
% less_numeral_extra(3)
tff(fact_2334_zero__less__iff__neq__zero,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
<=> ( Nb != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
tff(fact_2335_gr__implies__not__zero,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( Nb != zero_zero_nat ) ) ).
% gr_implies_not_zero
tff(fact_2336_not__less__zero,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).
% not_less_zero
tff(fact_2337_gr__zeroI,axiom,
! [Nb: nat] :
( ( Nb != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% gr_zeroI
tff(fact_2338_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).
% VEBT_internal.membermima.simps(2)
tff(fact_2339_zero__neq__numeral,axiom,
! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).
% zero_neq_numeral
tff(fact_2340_zero__neq__numeral,axiom,
! [Nb: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(Nb) ).
% zero_neq_numeral
tff(fact_2341_zero__neq__numeral,axiom,
! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).
% zero_neq_numeral
tff(fact_2342_zero__neq__numeral,axiom,
! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).
% zero_neq_numeral
tff(fact_2343_zero__neq__numeral,axiom,
! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).
% zero_neq_numeral
tff(fact_2344_zero__neq__numeral,axiom,
! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).
% zero_neq_numeral
tff(fact_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_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_2356_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_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_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_2369_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_2370_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
tff(fact_2371_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
tff(fact_2372_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
tff(fact_2373_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
tff(fact_2374_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
tff(fact_2375_verit__sum__simplify,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).
% verit_sum_simplify
tff(fact_2376_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_2377_verit__sum__simplify,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).
% verit_sum_simplify
tff(fact_2378_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_2379_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_2380_add_Ogroup__left__neutral,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).
% add.group_left_neutral
tff(fact_2381_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_2382_add_Ogroup__left__neutral,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).
% add.group_left_neutral
tff(fact_2383_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_2384_add_Ocomm__neutral,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).
% add.comm_neutral
tff(fact_2385_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_2386_add_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).
% add.comm_neutral
tff(fact_2387_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_2388_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_2389_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_2390_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_2391_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2392_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_2393_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_2394_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_2395_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_2396_eq__iff__diff__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).
% eq_iff_diff_eq_0
tff(fact_2397_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_2398_power__not__zero,axiom,
! [A: rat,Nb: nat] :
( ( A != zero_zero_rat )
=> ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).
% power_not_zero
tff(fact_2399_power__not__zero,axiom,
! [A: nat,Nb: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).
% power_not_zero
tff(fact_2400_power__not__zero,axiom,
! [A: real,Nb: nat] :
( ( A != zero_zero_real )
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).
% power_not_zero
tff(fact_2401_power__not__zero,axiom,
! [A: int,Nb: nat] :
( ( A != zero_zero_int )
=> ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).
% power_not_zero
tff(fact_2402_power__not__zero,axiom,
! [A: complex,Nb: nat] :
( ( A != zero_zero_complex )
=> ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).
% power_not_zero
tff(fact_2403_num_Osize_I4_J,axiom,
aa_num_nat(size_size_num,one) = zero_zero_nat ).
% num.size(4)
tff(fact_2404_not0__implies__Suc,axiom,
! [Nb: nat] :
( ( Nb != zero_zero_nat )
=> ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).
% not0_implies_Suc
tff(fact_2405_Zero__not__Suc,axiom,
! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).
% Zero_not_Suc
tff(fact_2406_Zero__neq__Suc,axiom,
! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).
% Zero_neq_Suc
tff(fact_2407_Suc__neq__Zero,axiom,
! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).
% Suc_neq_Zero
tff(fact_2408_zero__induct,axiom,
! [P: fun_nat_o,K: nat] :
( aa_nat_o(P,K)
=> ( ! [N: nat] :
( aa_nat_o(P,aa_nat_nat(suc,N))
=> aa_nat_o(P,N) )
=> aa_nat_o(P,zero_zero_nat) ) ) ).
% zero_induct
tff(fact_2409_diff__induct,axiom,
! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X4),zero_zero_nat)
=> ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
=> ( ! [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,X4),Y3)
=> aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3)) )
=> aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).
% diff_induct
tff(fact_2410_nat__induct,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,zero_zero_nat)
=> ( ! [N: nat] :
( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(suc,N)) )
=> aa_nat_o(P,Nb) ) ) ).
% nat_induct
tff(fact_2411_vebt__buildup_Ocases,axiom,
! [Xb: nat] :
( ( Xb != zero_zero_nat )
=> ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
=> ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).
% vebt_buildup.cases
tff(fact_2412_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).
% old.nat.exhaust
tff(fact_2413_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat = aa_nat_nat(suc,X2) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
tff(fact_2414_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).
% old.nat.distinct(1)
tff(fact_2415_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).
% old.nat.distinct(2)
tff(fact_2416_nat_Odistinct_I1_J,axiom,
! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).
% nat.distinct(1)
tff(fact_2417_infinite__descent0,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,zero_zero_nat)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( ~ aa_nat_o(P,N)
=> ? [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
& ~ aa_nat_o(P,M3) ) ) )
=> aa_nat_o(P,Nb) ) ) ).
% infinite_descent0
tff(fact_2418_gr__implies__not0,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( Nb != zero_zero_nat ) ) ).
% gr_implies_not0
tff(fact_2419_less__zeroE,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).
% less_zeroE
tff(fact_2420_not__less0,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).
% not_less0
tff(fact_2421_not__gr0,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
<=> ( Nb = zero_zero_nat ) ) ).
% not_gr0
tff(fact_2422_gr0I,axiom,
! [Nb: nat] :
( ( Nb != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% gr0I
tff(fact_2423_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).
% bot_nat_0.extremum_strict
tff(fact_2424_le__0__eq,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
<=> ( Nb = zero_zero_nat ) ) ).
% le_0_eq
tff(fact_2425_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
tff(fact_2426_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
<=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
tff(fact_2427_less__eq__nat_Osimps_I1_J,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).
% less_eq_nat.simps(1)
tff(fact_2428_add__eq__self__zero,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
=> ( Nb = zero_zero_nat ) ) ).
% add_eq_self_zero
tff(fact_2429_plus__nat_Oadd__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).
% plus_nat.add_0
tff(fact_2430_diffs0__imp__equal,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
=> ( Mb = Nb ) ) ) ).
% diffs0_imp_equal
tff(fact_2431_minus__nat_Odiff__0,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).
% minus_nat.diff_0
tff(fact_2432_nat__mult__eq__cancel__disj,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
<=> ( ( K = zero_zero_nat )
| ( Mb = Nb ) ) ) ).
% nat_mult_eq_cancel_disj
tff(fact_2433_mult__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).
% mult_0
tff(fact_2434_power__eq__iff__eq__base,axiom,
! [Nb: nat,A: rat,B: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2435_power__eq__iff__eq__base,axiom,
! [Nb: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2436_power__eq__iff__eq__base,axiom,
! [Nb: nat,A: int,B: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2437_power__eq__iff__eq__base,axiom,
! [Nb: nat,A: real,B: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2438_power__eq__imp__eq__base,axiom,
! [A: rat,Nb: nat,B: rat] :
( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2439_power__eq__imp__eq__base,axiom,
! [A: nat,Nb: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2440_power__eq__imp__eq__base,axiom,
! [A: int,Nb: nat,B: int] :
( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2441_power__eq__imp__eq__base,axiom,
! [A: real,Nb: nat,B: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2442_lambda__zero,axiom,
aTP_Lamm_an = times_times_complex(zero_zero_complex) ).
% lambda_zero
tff(fact_2443_lambda__zero,axiom,
aTP_Lamm_ao = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).
% lambda_zero
tff(fact_2444_lambda__zero,axiom,
aTP_Lamm_ap = times_times_rat(zero_zero_rat) ).
% lambda_zero
tff(fact_2445_lambda__zero,axiom,
aTP_Lamm_aq = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).
% lambda_zero
tff(fact_2446_lambda__zero,axiom,
aTP_Lamm_ar = aa_int_fun_int_int(times_times_int,zero_zero_int) ).
% lambda_zero
tff(fact_2447_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,Va,Vb)),Xb)
<=> ( ( Xb = Mia )
| ( Xb = Maa ) ) ) ).
% VEBT_internal.membermima.simps(3)
tff(fact_2448_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd)),Ve) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_2449_power__strict__mono,axiom,
! [A: rat,B: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).
% power_strict_mono
tff(fact_2450_power__strict__mono,axiom,
! [A: nat,B: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).
% power_strict_mono
tff(fact_2451_power__strict__mono,axiom,
! [A: int,B: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).
% power_strict_mono
tff(fact_2452_power__strict__mono,axiom,
! [A: real,B: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).
% power_strict_mono
tff(fact_2453_not__numeral__le__zero,axiom,
! [Nb: num] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Nb)),zero_z5237406670263579293d_enat) ).
% not_numeral_le_zero
tff(fact_2454_not__numeral__le__zero,axiom,
! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).
% not_numeral_le_zero
tff(fact_2455_not__numeral__le__zero,axiom,
! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).
% not_numeral_le_zero
tff(fact_2456_not__numeral__le__zero,axiom,
! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).
% not_numeral_le_zero
tff(fact_2457_not__numeral__le__zero,axiom,
! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).
% not_numeral_le_zero
tff(fact_2458_zero__le__numeral,axiom,
! [Nb: num] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Nb)) ).
% zero_le_numeral
tff(fact_2459_zero__le__numeral,axiom,
! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).
% zero_le_numeral
tff(fact_2460_zero__le__numeral,axiom,
! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).
% zero_le_numeral
tff(fact_2461_zero__le__numeral,axiom,
! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).
% zero_le_numeral
tff(fact_2462_zero__le__numeral,axiom,
! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).
% zero_le_numeral
tff(fact_2463_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).
% mult_mono
tff(fact_2464_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).
% mult_mono
tff(fact_2465_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2466_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2467_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).
% mult_mono'
tff(fact_2468_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).
% mult_mono'
tff(fact_2469_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2470_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2471_zero__le__square,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).
% zero_le_square
tff(fact_2472_zero__le__square,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(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_2473_zero__le__square,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).
% zero_le_square
tff(fact_2474_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).
% split_mult_pos_le
tff(fact_2475_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).
% split_mult_pos_le
tff(fact_2476_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).
% split_mult_pos_le
tff(fact_2477_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(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_2478_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2479_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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_2480_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).
% mult_nonpos_nonpos
tff(fact_2481_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_nonpos_nonpos
tff(fact_2482_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).
% mult_nonpos_nonpos
tff(fact_2483_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2484_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).
% mult_left_mono
tff(fact_2485_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2486_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2487_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(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_2488_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2489_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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_2490_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2491_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% mult_right_mono
tff(fact_2492_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2493_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2494_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).
% mult_le_0_iff
tff(fact_2495_mult__le__0__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).
% mult_le_0_iff
tff(fact_2496_mult__le__0__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).
% mult_le_0_iff
tff(fact_2497_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).
% split_mult_neg_le
tff(fact_2498_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
=> aa_nat_o(aa_nat_fun_nat_o(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_2499_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
=> aa_int_o(aa_int_fun_int_o(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_2500_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).
% split_mult_neg_le
tff(fact_2501_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).
% mult_nonneg_nonneg
tff(fact_2502_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_2503_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(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_2504_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).
% mult_nonneg_nonneg
tff(fact_2505_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).
% mult_nonneg_nonpos
tff(fact_2506_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(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_2507_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2508_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).
% mult_nonneg_nonpos
tff(fact_2509_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).
% mult_nonpos_nonneg
tff(fact_2510_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_2511_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(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_2512_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).
% mult_nonpos_nonneg
tff(fact_2513_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2514_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(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_2515_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2516_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2517_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).
% zero_le_mult_iff
tff(fact_2518_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).
% zero_le_mult_iff
tff(fact_2519_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).
% zero_le_mult_iff
tff(fact_2520_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2521_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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_2522_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2523_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2524_not__numeral__less__zero,axiom,
! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).
% not_numeral_less_zero
tff(fact_2525_not__numeral__less__zero,axiom,
! [Nb: num] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Nb)),zero_z5237406670263579293d_enat) ).
% not_numeral_less_zero
tff(fact_2526_not__numeral__less__zero,axiom,
! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).
% not_numeral_less_zero
tff(fact_2527_not__numeral__less__zero,axiom,
! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).
% not_numeral_less_zero
tff(fact_2528_not__numeral__less__zero,axiom,
! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).
% not_numeral_less_zero
tff(fact_2529_zero__less__numeral,axiom,
! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).
% zero_less_numeral
tff(fact_2530_zero__less__numeral,axiom,
! [Nb: num] : aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Nb)) ).
% zero_less_numeral
tff(fact_2531_zero__less__numeral,axiom,
! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).
% zero_less_numeral
tff(fact_2532_zero__less__numeral,axiom,
! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).
% zero_less_numeral
tff(fact_2533_zero__less__numeral,axiom,
! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).
% zero_less_numeral
tff(fact_2534_not__one__le__zero,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).
% not_one_le_zero
tff(fact_2535_not__one__le__zero,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).
% not_one_le_zero
tff(fact_2536_not__one__le__zero,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).
% not_one_le_zero
tff(fact_2537_not__one__le__zero,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).
% not_one_le_zero
tff(fact_2538_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2539_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2540_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2541_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2542_zero__less__one__class_Ozero__le__one,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).
% zero_less_one_class.zero_le_one
tff(fact_2543_zero__less__one__class_Ozero__le__one,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).
% zero_less_one_class.zero_le_one
tff(fact_2544_zero__less__one__class_Ozero__le__one,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).
% zero_less_one_class.zero_le_one
tff(fact_2545_zero__less__one__class_Ozero__le__one,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).
% zero_less_one_class.zero_le_one
tff(fact_2546_add__nonpos__eq__0__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
=> ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
<=> ( ( Xb = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2547_add__nonpos__eq__0__iff,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
<=> ( ( Xb = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2548_add__nonpos__eq__0__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
<=> ( ( Xb = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2549_add__nonpos__eq__0__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
<=> ( ( Xb = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2550_add__nonneg__eq__0__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
<=> ( ( Xb = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2551_add__nonneg__eq__0__iff,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
<=> ( ( Xb = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2552_add__nonneg__eq__0__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
<=> ( ( Xb = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2553_add__nonneg__eq__0__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
<=> ( ( Xb = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2554_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).
% add_nonpos_nonpos
tff(fact_2555_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(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_2556_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2557_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).
% add_nonpos_nonpos
tff(fact_2558_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).
% add_nonneg_nonneg
tff(fact_2559_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_2560_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(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_2561_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% add_nonneg_nonneg
tff(fact_2562_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).
% add_increasing2
tff(fact_2563_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_increasing2
tff(fact_2564_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_increasing2
tff(fact_2565_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).
% add_increasing2
tff(fact_2566_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).
% add_decreasing2
tff(fact_2567_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_2568_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_2569_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_2570_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).
% add_increasing
tff(fact_2571_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_increasing
tff(fact_2572_add__increasing,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_increasing
tff(fact_2573_add__increasing,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).
% add_increasing
tff(fact_2574_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).
% add_decreasing
tff(fact_2575_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).
% add_decreasing
tff(fact_2576_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).
% add_decreasing
tff(fact_2577_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).
% add_decreasing
tff(fact_2578_mult__neg__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).
% mult_neg_neg
tff(fact_2579_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).
% mult_neg_neg
tff(fact_2580_mult__neg__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_neg_neg
tff(fact_2581_not__square__less__zero,axiom,
! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).
% not_square_less_zero
tff(fact_2582_not__square__less__zero,axiom,
! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).
% not_square_less_zero
tff(fact_2583_not__square__less__zero,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(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_2584_mult__less__0__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).
% mult_less_0_iff
tff(fact_2585_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).
% mult_less_0_iff
tff(fact_2586_mult__less__0__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).
% mult_less_0_iff
tff(fact_2587_mult__neg__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).
% mult_neg_pos
tff(fact_2588_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).
% mult_neg_pos
tff(fact_2589_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).
% mult_neg_pos
tff(fact_2590_mult__neg__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).
% mult_neg_pos
tff(fact_2591_mult__pos__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).
% mult_pos_neg
tff(fact_2592_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).
% mult_pos_neg
tff(fact_2593_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).
% mult_pos_neg
tff(fact_2594_mult__pos__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).
% mult_pos_neg
tff(fact_2595_mult__pos__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2596_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).
% mult_pos_pos
tff(fact_2597_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2598_mult__pos__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2599_mult__pos__neg2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).
% mult_pos_neg2
tff(fact_2600_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).
% mult_pos_neg2
tff(fact_2601_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).
% mult_pos_neg2
tff(fact_2602_mult__pos__neg2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).
% mult_pos_neg2
tff(fact_2603_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).
% zero_less_mult_iff
tff(fact_2604_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).
% zero_less_mult_iff
tff(fact_2605_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).
% zero_less_mult_iff
tff(fact_2606_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).
% zero_less_mult_pos
tff(fact_2607_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).
% zero_less_mult_pos
tff(fact_2608_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).
% zero_less_mult_pos
tff(fact_2609_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).
% zero_less_mult_pos
tff(fact_2610_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2611_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2612_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2613_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2614_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2615_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2616_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2617_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2618_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2619_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> ( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2620_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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_2621_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(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_2622_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2623_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2624_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2625_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(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_2626_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2627_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2628_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2629_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
& aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2630_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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_2631_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(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_2632_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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_2633_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2634_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2635_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(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_2636_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2637_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2638_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2639_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
& aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2640_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2641_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2642_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(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_2643_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2644_not__one__less__zero,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).
% not_one_less_zero
tff(fact_2645_not__one__less__zero,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).
% not_one_less_zero
tff(fact_2646_not__one__less__zero,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).
% not_one_less_zero
tff(fact_2647_not__one__less__zero,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).
% not_one_less_zero
tff(fact_2648_zero__less__one,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).
% zero_less_one
tff(fact_2649_zero__less__one,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).
% zero_less_one
tff(fact_2650_zero__less__one,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).
% zero_less_one
tff(fact_2651_zero__less__one,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).
% zero_less_one
tff(fact_2652_less__numeral__extra_I1_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).
% less_numeral_extra(1)
tff(fact_2653_less__numeral__extra_I1_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).
% less_numeral_extra(1)
tff(fact_2654_less__numeral__extra_I1_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).
% less_numeral_extra(1)
tff(fact_2655_less__numeral__extra_I1_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).
% less_numeral_extra(1)
tff(fact_2656_add__less__zeroD,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
| aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).
% add_less_zeroD
tff(fact_2657_add__less__zeroD,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Y)),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).
% add_less_zeroD
tff(fact_2658_add__less__zeroD,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).
% add_less_zeroD
tff(fact_2659_add__neg__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).
% add_neg_neg
tff(fact_2660_add__neg__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).
% add_neg_neg
tff(fact_2661_add__neg__neg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_neg_neg
tff(fact_2662_add__neg__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_neg_neg
tff(fact_2663_add__pos__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% add_pos_pos
tff(fact_2664_add__pos__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).
% add_pos_pos
tff(fact_2665_add__pos__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_pos_pos
tff(fact_2666_add__pos__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_pos_pos
tff(fact_2667_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ ! [C3: nat] :
( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_2668_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).
% pos_add_strict
tff(fact_2669_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).
% pos_add_strict
tff(fact_2670_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% pos_add_strict
tff(fact_2671_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% pos_add_strict
tff(fact_2672_le__iff__diff__le__0,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).
% le_iff_diff_le_0
tff(fact_2673_le__iff__diff__le__0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_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_2674_le__iff__diff__le__0,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
<=> aa_real_o(aa_real_fun_real_o(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_2675_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).
% divide_le_0_iff
tff(fact_2676_divide__le__0__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).
% divide_le_0_iff
tff(fact_2677_divide__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).
% divide_right_mono
tff(fact_2678_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2679_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).
% zero_le_divide_iff
tff(fact_2680_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).
% zero_le_divide_iff
tff(fact_2681_divide__nonneg__nonneg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% divide_nonneg_nonneg
tff(fact_2682_divide__nonneg__nonneg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% divide_nonneg_nonneg
tff(fact_2683_divide__nonneg__nonpos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).
% divide_nonneg_nonpos
tff(fact_2684_divide__nonneg__nonpos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).
% divide_nonneg_nonpos
tff(fact_2685_divide__nonpos__nonneg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).
% divide_nonpos_nonneg
tff(fact_2686_divide__nonpos__nonneg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).
% divide_nonpos_nonneg
tff(fact_2687_divide__nonpos__nonpos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% divide_nonpos_nonpos
tff(fact_2688_divide__nonpos__nonpos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% divide_nonpos_nonpos
tff(fact_2689_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).
% divide_right_mono_neg
tff(fact_2690_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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_2691_less__iff__diff__less__0,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).
% less_iff_diff_less_0
tff(fact_2692_less__iff__diff__less__0,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).
% less_iff_diff_less_0
tff(fact_2693_less__iff__diff__less__0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).
% less_iff_diff_less_0
tff(fact_2694_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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_2695_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).
% divide_strict_right_mono_neg
tff(fact_2696_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2697_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).
% divide_strict_right_mono
tff(fact_2698_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).
% zero_less_divide_iff
tff(fact_2699_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).
% zero_less_divide_iff
tff(fact_2700_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
tff(fact_2701_divide__less__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
tff(fact_2702_divide__less__0__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).
% divide_less_0_iff
tff(fact_2703_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).
% divide_less_0_iff
tff(fact_2704_divide__pos__pos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% divide_pos_pos
tff(fact_2705_divide__pos__pos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% divide_pos_pos
tff(fact_2706_divide__pos__neg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).
% divide_pos_neg
tff(fact_2707_divide__pos__neg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).
% divide_pos_neg
tff(fact_2708_divide__neg__pos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).
% divide_neg_pos
tff(fact_2709_divide__neg__pos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).
% divide_neg_pos
tff(fact_2710_divide__neg__neg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% divide_neg_neg
tff(fact_2711_divide__neg__neg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% divide_neg_neg
tff(fact_2712_power__mono,axiom,
! [A: rat,B: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).
% power_mono
tff(fact_2713_power__mono,axiom,
! [A: nat,B: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).
% power_mono
tff(fact_2714_power__mono,axiom,
! [A: int,B: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).
% power_mono
tff(fact_2715_power__mono,axiom,
! [A: real,B: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).
% power_mono
tff(fact_2716_zero__le__power,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).
% zero_le_power
tff(fact_2717_zero__le__power,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).
% zero_le_power
tff(fact_2718_zero__le__power,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).
% zero_le_power
tff(fact_2719_zero__le__power,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).
% zero_le_power
tff(fact_2720_zero__less__power,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).
% zero_less_power
tff(fact_2721_zero__less__power,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).
% zero_less_power
tff(fact_2722_zero__less__power,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).
% zero_less_power
tff(fact_2723_zero__less__power,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).
% zero_less_power
tff(fact_2724_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_2725_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_2726_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
<=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_2727_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_2728_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_2729_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
<=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_2730_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_2731_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_2732_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(times_times_rat(A),C) = B )
=> ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).
% eq_divide_imp
tff(fact_2733_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_2734_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_2735_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B = aa_rat_rat(times_times_rat(A),C) )
=> ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).
% divide_eq_imp
tff(fact_2736_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).
% eq_divide_eq
tff(fact_2737_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).
% eq_divide_eq
tff(fact_2738_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = aa_rat_rat(divide_divide_rat(B),C) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).
% eq_divide_eq
tff(fact_2739_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
<=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).
% divide_eq_eq
tff(fact_2740_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
<=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).
% divide_eq_eq
tff(fact_2741_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( aa_rat_rat(divide_divide_rat(B),C) = A )
<=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).
% divide_eq_eq
tff(fact_2742_frac__eq__eq,axiom,
! [Y: complex,Z: complex,Xb: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
<=> ( aa_complex_complex(times_times_complex(Xb),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_2743_frac__eq__eq,axiom,
! [Y: real,Z: real,Xb: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_2744_frac__eq__eq,axiom,
! [Y: rat,Z: rat,Xb: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( aa_rat_rat(divide_divide_rat(Xb),Y) = aa_rat_rat(divide_divide_rat(W),Z) )
<=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_2745_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_2746_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_2747_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
<=> ( A = B ) ) ) ).
% right_inverse_eq
tff(fact_2748_power__0,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).
% power_0
tff(fact_2749_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_2750_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_2751_power__0,axiom,
! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).
% power_0
tff(fact_2752_power__0,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).
% power_0
tff(fact_2753_less__Suc__eq__0__disj,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
<=> ( ( Mb = zero_zero_nat )
| ? [J3: nat] :
( ( Mb = aa_nat_nat(suc,J3) )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).
% less_Suc_eq_0_disj
tff(fact_2754_gr0__implies__Suc,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).
% gr0_implies_Suc
tff(fact_2755_All__less__Suc2,axiom,
! [Nb: nat,P: fun_nat_o] :
( ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
=> aa_nat_o(P,I4) )
<=> ( aa_nat_o(P,zero_zero_nat)
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
=> aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).
% All_less_Suc2
tff(fact_2756_gr0__conv__Suc,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
<=> ? [M4: nat] : Nb = aa_nat_nat(suc,M4) ) ).
% gr0_conv_Suc
tff(fact_2757_Ex__less__Suc2,axiom,
! [Nb: nat,P: fun_nat_o] :
( ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
& aa_nat_o(P,I4) )
<=> ( aa_nat_o(P,zero_zero_nat)
| ? [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
& aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).
% Ex_less_Suc2
tff(fact_2758_add__is__1,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
& ( Nb = zero_zero_nat ) )
| ( ( Mb = zero_zero_nat )
& ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
% add_is_1
tff(fact_2759_one__is__add,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
<=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
& ( Nb = zero_zero_nat ) )
| ( ( Mb = zero_zero_nat )
& ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
% one_is_add
tff(fact_2760_option_Osize_I4_J,axiom,
! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(4)
tff(fact_2761_option_Osize_I4_J,axiom,
! [X2: nat] : size_size_option_nat(some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(4)
tff(fact_2762_option_Osize_I4_J,axiom,
! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(4)
tff(fact_2763_One__nat__def,axiom,
one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).
% One_nat_def
tff(fact_2764_ex__least__nat__le,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,Nb)
=> ( ~ aa_nat_o(P,zero_zero_nat)
=> ? [K3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Nb)
& ! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K3)
=> ~ aa_nat_o(P,I2) )
& aa_nat_o(P,K3) ) ) ) ).
% ex_least_nat_le
tff(fact_2765_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ? [K3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_2766_option_Osize_I3_J,axiom,
size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(3)
tff(fact_2767_option_Osize_I3_J,axiom,
size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(3)
tff(fact_2768_option_Osize_I3_J,axiom,
size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(3)
tff(fact_2769_diff__less,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).
% diff_less
tff(fact_2770_nat__mult__less__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% nat_mult_less_cancel1
tff(fact_2771_nat__mult__eq__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
<=> ( Mb = Nb ) ) ) ).
% nat_mult_eq_cancel1
tff(fact_2772_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).
% mult_less_mono2
tff(fact_2773_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).
% mult_less_mono1
tff(fact_2774_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| ( Nb = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
tff(fact_2775_diff__add__0,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).
% diff_add_0
tff(fact_2776_mult__eq__self__implies__10,axiom,
! [Mb: nat,Nb: nat] :
( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
=> ( ( Nb = one_one_nat )
| ( Mb = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
tff(fact_2777_nat__power__less__imp__less,axiom,
! [I: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).
% nat_power_less_imp_less
tff(fact_2778_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) ).
% vebt_member.simps(3)
tff(fact_2779_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)),Vi) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_2780_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S2),Xb) = vEBT_Node(Info,zero_zero_nat,Ts,S2) ).
% vebt_insert.simps(2)
tff(fact_2781_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).
% mult_le_cancel_left
tff(fact_2782_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).
% mult_le_cancel_left
tff(fact_2783_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).
% mult_le_cancel_left
tff(fact_2784_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).
% mult_le_cancel_right
tff(fact_2785_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).
% mult_le_cancel_right
tff(fact_2786_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).
% mult_le_cancel_right
tff(fact_2787_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2788_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2789_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2790_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2791_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).
% mult_strict_mono
tff(fact_2792_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(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_2793_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2794_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2795_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% mult_less_cancel_left
tff(fact_2796_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_left
tff(fact_2797_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% mult_less_cancel_left
tff(fact_2798_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2799_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2800_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2801_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2802_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2803_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(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_2804_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2805_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2806_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% mult_less_cancel_right
tff(fact_2807_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_right
tff(fact_2808_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% mult_less_cancel_right
tff(fact_2809_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2810_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2811_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2812_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2813_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> ( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2814_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2815_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2816_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2817_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2818_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2819_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2820_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2821_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2822_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2823_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2824_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(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_2825_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2826_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2827_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(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_2828_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_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_2829_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(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_2830_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(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_2831_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).
% add_strict_increasing2
tff(fact_2832_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_2833_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_2834_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_2835_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).
% add_strict_increasing
tff(fact_2836_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_strict_increasing
tff(fact_2837_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_strict_increasing
tff(fact_2838_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).
% add_strict_increasing
tff(fact_2839_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).
% add_pos_nonneg
tff(fact_2840_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_2841_add__pos__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_2842_add__pos__nonneg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_2843_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).
% add_nonpos_neg
tff(fact_2844_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_nonpos_neg
tff(fact_2845_add__nonpos__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_nonpos_neg
tff(fact_2846_add__nonpos__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).
% add_nonpos_neg
tff(fact_2847_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).
% add_nonneg_pos
tff(fact_2848_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_2849_add__nonneg__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_2850_add__nonneg__pos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_2851_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).
% add_neg_nonpos
tff(fact_2852_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_neg_nonpos
tff(fact_2853_add__neg__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_neg_nonpos
tff(fact_2854_add__neg__nonpos,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).
% add_neg_nonpos
tff(fact_2855_field__le__epsilon,axiom,
! [Xb: rat,Y: rat] :
( ! [E2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(Y),E2)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).
% field_le_epsilon
tff(fact_2856_field__le__epsilon,axiom,
! [Xb: real,Y: real] :
( ! [E2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).
% field_le_epsilon
tff(fact_2857_mult__left__le,axiom,
! [C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).
% mult_left_le
tff(fact_2858_mult__left__le,axiom,
! [C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).
% mult_left_le
tff(fact_2859_mult__left__le,axiom,
! [C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).
% mult_left_le
tff(fact_2860_mult__left__le,axiom,
! [C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).
% mult_left_le
tff(fact_2861_mult__le__one,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).
% mult_le_one
tff(fact_2862_mult__le__one,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(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_2863_mult__le__one,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(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_2864_mult__le__one,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).
% mult_le_one
tff(fact_2865_mult__right__le__one__le,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Y)),Xb) ) ) ) ).
% mult_right_le_one_le
tff(fact_2866_mult__right__le__one__le,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),Xb) ) ) ) ).
% mult_right_le_one_le
tff(fact_2867_mult__right__le__one__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),Xb) ) ) ) ).
% mult_right_le_one_le
tff(fact_2868_mult__left__le__one__le,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),Xb)),Xb) ) ) ) ).
% mult_left_le_one_le
tff(fact_2869_mult__left__le__one__le,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)),Xb) ) ) ) ).
% mult_left_le_one_le
tff(fact_2870_mult__left__le__one__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)),Xb) ) ) ) ).
% mult_left_le_one_le
tff(fact_2871_frac__le,axiom,
! [Y: rat,Xb: rat,W: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).
% frac_le
tff(fact_2872_frac__le,axiom,
! [Y: real,Xb: real,W: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).
% frac_le
tff(fact_2873_frac__less,axiom,
! [Xb: rat,Y: rat,W: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).
% frac_less
tff(fact_2874_frac__less,axiom,
! [Xb: real,Y: real,W: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).
% frac_less
tff(fact_2875_frac__less2,axiom,
! [Xb: rat,Y: rat,W: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).
% frac_less2
tff(fact_2876_frac__less2,axiom,
! [Xb: real,Y: real,W: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).
% frac_less2
tff(fact_2877_divide__le__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).
% divide_le_cancel
tff(fact_2878_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).
% divide_le_cancel
tff(fact_2879_divide__nonneg__neg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).
% divide_nonneg_neg
tff(fact_2880_divide__nonneg__neg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).
% divide_nonneg_neg
tff(fact_2881_divide__nonneg__pos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% divide_nonneg_pos
tff(fact_2882_divide__nonneg__pos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% divide_nonneg_pos
tff(fact_2883_divide__nonpos__neg,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% divide_nonpos_neg
tff(fact_2884_divide__nonpos__neg,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% divide_nonpos_neg
tff(fact_2885_divide__nonpos__pos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).
% divide_nonpos_pos
tff(fact_2886_divide__nonpos__pos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).
% divide_nonpos_pos
tff(fact_2887_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_2888_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2889_div__positive,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_positive
tff(fact_2890_div__positive,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)) ) ) ).
% div_positive
tff(fact_2891_sum__squares__ge__zero,axiom,
! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))) ).
% sum_squares_ge_zero
tff(fact_2892_sum__squares__ge__zero,axiom,
! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).
% sum_squares_ge_zero
tff(fact_2893_sum__squares__ge__zero,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).
% sum_squares_ge_zero
tff(fact_2894_sum__squares__le__zero__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
<=> ( ( Xb = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2895_sum__squares__le__zero__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
<=> ( ( Xb = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2896_sum__squares__le__zero__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
<=> ( ( Xb = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2897_not__sum__squares__lt__zero,axiom,
! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).
% not_sum_squares_lt_zero
tff(fact_2898_not__sum__squares__lt__zero,axiom,
! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).
% not_sum_squares_lt_zero
tff(fact_2899_not__sum__squares__lt__zero,axiom,
! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).
% not_sum_squares_lt_zero
tff(fact_2900_sum__squares__gt__zero__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
<=> ( ( Xb != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2901_sum__squares__gt__zero__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)))
<=> ( ( Xb != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2902_sum__squares__gt__zero__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
<=> ( ( Xb != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2903_zero__less__two,axiom,
aa_real_o(aa_real_fun_real_o(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_2904_zero__less__two,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).
% zero_less_two
tff(fact_2905_zero__less__two,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).
% zero_less_two
tff(fact_2906_zero__less__two,axiom,
aa_int_o(aa_int_fun_int_o(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_2907_power__less__imp__less__base,axiom,
! [A: rat,Nb: nat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).
% power_less_imp_less_base
tff(fact_2908_power__less__imp__less__base,axiom,
! [A: nat,Nb: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% power_less_imp_less_base
tff(fact_2909_power__less__imp__less__base,axiom,
! [A: int,Nb: nat,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% power_less_imp_less_base
tff(fact_2910_power__less__imp__less__base,axiom,
! [A: real,Nb: nat,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% power_less_imp_less_base
tff(fact_2911_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_2912_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2913_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).
% divide_less_eq
tff(fact_2914_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).
% divide_less_eq
tff(fact_2915_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).
% less_divide_eq
tff(fact_2916_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).
% less_divide_eq
tff(fact_2917_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).
% neg_divide_less_eq
tff(fact_2918_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).
% neg_divide_less_eq
tff(fact_2919_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).
% neg_less_divide_eq
tff(fact_2920_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% neg_less_divide_eq
tff(fact_2921_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).
% pos_divide_less_eq
tff(fact_2922_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% pos_divide_less_eq
tff(fact_2923_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).
% pos_less_divide_eq
tff(fact_2924_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).
% pos_less_divide_eq
tff(fact_2925_mult__imp__div__pos__less,axiom,
! [Y: real,Xb: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).
% mult_imp_div_pos_less
tff(fact_2926_mult__imp__div__pos__less,axiom,
! [Y: rat,Xb: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Z) ) ) ).
% mult_imp_div_pos_less
tff(fact_2927_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% mult_imp_less_div_pos
tff(fact_2928_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% mult_imp_less_div_pos
tff(fact_2929_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
=> aa_real_o(aa_real_fun_real_o(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_2930_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).
% divide_strict_left_mono
tff(fact_2931_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
=> aa_real_o(aa_real_fun_real_o(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_2932_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_2933_less__divide__eq__1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% less_divide_eq_1
tff(fact_2934_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% less_divide_eq_1
tff(fact_2935_divide__less__eq__1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
tff(fact_2936_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
tff(fact_2937_power__le__one,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).
% power_le_one
tff(fact_2938_power__le__one,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).
% power_le_one
tff(fact_2939_power__le__one,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),one_one_int) ) ) ).
% power_le_one
tff(fact_2940_power__le__one,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).
% power_le_one
tff(fact_2941_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: complex,C: complex] :
( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2942_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2943_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2944_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W: num] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
<=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2945_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W) )
<=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2946_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
<=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2947_power__le__imp__le__base,axiom,
! [A: rat,Nb: nat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).
% power_le_imp_le_base
tff(fact_2948_power__le__imp__le__base,axiom,
! [A: nat,Nb: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% power_le_imp_le_base
tff(fact_2949_power__le__imp__le__base,axiom,
! [A: int,Nb: nat,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% power_le_imp_le_base
tff(fact_2950_power__le__imp__le__base,axiom,
! [A: real,Nb: nat,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% power_le_imp_le_base
tff(fact_2951_power__inject__base,axiom,
! [A: rat,Nb: nat,B: rat] :
( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2952_power__inject__base,axiom,
! [A: nat,Nb: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2953_power__inject__base,axiom,
! [A: int,Nb: nat,B: int] :
( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2954_power__inject__base,axiom,
! [A: real,Nb: nat,B: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2955_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).
% vebt_member.simps(4)
tff(fact_2956_add__divide__eq__if__simps_I2_J,axiom,
! [A: complex,Z: complex,B: complex] :
aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,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_2957_add__divide__eq__if__simps_I2_J,axiom,
! [A: real,Z: real,B: real] :
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,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_2958_add__divide__eq__if__simps_I2_J,axiom,
! [A: rat,Z: rat,B: rat] :
aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).
% add_divide_eq_if_simps(2)
tff(fact_2959_add__divide__eq__if__simps_I1_J,axiom,
! [A: complex,B: complex,Z: complex] :
aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,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_2960_add__divide__eq__if__simps_I1_J,axiom,
! [A: real,B: real,Z: real] :
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,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_2961_add__divide__eq__if__simps_I1_J,axiom,
! [A: rat,B: rat,Z: rat] :
aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).
% add_divide_eq_if_simps(1)
tff(fact_2962_add__frac__eq,axiom,
! [Y: complex,Z: complex,Xb: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_2963_add__frac__eq,axiom,
! [Y: real,Z: real,Xb: 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,Xb),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,Xb),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_2964_add__frac__eq,axiom,
! [Y: rat,Z: rat,Xb: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_2965_add__frac__num,axiom,
! [Y: complex,Xb: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).
% add_frac_num
tff(fact_2966_add__frac__num,axiom,
! [Y: real,Xb: 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,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).
% add_frac_num
tff(fact_2967_add__frac__num,axiom,
! [Y: rat,Xb: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).
% add_frac_num
tff(fact_2968_add__num__frac,axiom,
! [Y: complex,Z: complex,Xb: complex] :
( ( Y != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).
% add_num_frac
tff(fact_2969_add__num__frac,axiom,
! [Y: real,Z: real,Xb: real] :
( ( Y != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).
% add_num_frac
tff(fact_2970_add__num__frac,axiom,
! [Y: rat,Z: rat,Xb: rat] :
( ( Y != zero_zero_rat )
=> ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xb),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).
% add_num_frac
tff(fact_2971_add__divide__eq__iff,axiom,
! [Z: complex,Xb: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Y)),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2972_add__divide__eq__iff,axiom,
! [Z: real,Xb: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),Z)),Y)),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2973_add__divide__eq__iff,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y)),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2974_divide__add__eq__iff,axiom,
! [Z: complex,Xb: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2975_divide__add__eq__iff,axiom,
! [Z: real,Xb: 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,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2976_divide__add__eq__iff,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2977_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_2978_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B)),one_one_int) ) ) ).
% div_add_self2
tff(fact_2979_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_2980_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_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(divide_divide_int(A),B)),one_one_int) ) ) ).
% div_add_self1
tff(fact_2981_divide__diff__eq__iff,axiom,
! [Z: complex,Xb: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_2982_divide__diff__eq__iff,axiom,
! [Z: real,Xb: 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,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_2983_divide__diff__eq__iff,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_2984_diff__divide__eq__iff,axiom,
! [Z: complex,Xb: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Y)),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_2985_diff__divide__eq__iff,axiom,
! [Z: real,Xb: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),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,Xb),Z)),Y)),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_2986_diff__divide__eq__iff,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y)),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_2987_diff__frac__eq,axiom,
! [Y: complex,Z: complex,Xb: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_2988_diff__frac__eq,axiom,
! [Y: real,Z: real,Xb: 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,Xb),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,Xb),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_2989_diff__frac__eq,axiom,
! [Y: rat,Z: rat,Xb: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_2990_add__divide__eq__if__simps_I4_J,axiom,
! [A: complex,B: complex,Z: complex] :
aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,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_2991_add__divide__eq__if__simps_I4_J,axiom,
! [A: real,B: real,Z: real] :
aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,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_2992_add__divide__eq__if__simps_I4_J,axiom,
! [A: rat,B: rat,Z: rat] :
aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).
% add_divide_eq_if_simps(4)
tff(fact_2993_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_2994_num_Osize_I5_J,axiom,
! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size(5)
tff(fact_2995_vebt__succ_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve) = none_nat ).
% vebt_succ.simps(4)
tff(fact_2996_ex__least__nat__less,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,Nb)
=> ( ~ aa_nat_o(P,zero_zero_nat)
=> ? [K3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
& ! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K3)
=> ~ aa_nat_o(P,I2) )
& aa_nat_o(P,aa_nat_nat(suc,K3)) ) ) ) ).
% ex_least_nat_less
tff(fact_2997_nat__induct__non__zero,axiom,
! [Nb: nat,P: fun_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(P,one_one_nat)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(suc,N)) ) )
=> aa_nat_o(P,Nb) ) ) ) ).
% nat_induct_non_zero
tff(fact_2998_diff__Suc__less,axiom,
! [Nb: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).
% diff_Suc_less
tff(fact_2999_one__less__mult,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).
% one_less_mult
tff(fact_3000_n__less__m__mult__n,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).
% n_less_m_mult_n
tff(fact_3001_n__less__n__mult__m,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).
% n_less_n_mult_m
tff(fact_3002_length__pos__if__in__set,axiom,
! [Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3003_length__pos__if__in__set,axiom,
! [Xb: complex,Xs: list_complex] :
( aa_set_complex_o(member_complex(Xb),set_complex2(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3004_length__pos__if__in__set,axiom,
! [Xb: real,Xs: list_real] :
( aa_set_real_o(member_real(Xb),set_real2(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3005_length__pos__if__in__set,axiom,
! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3006_length__pos__if__in__set,axiom,
! [Xb: $o,Xs: list_o] :
( member_o((Xb),set_o2(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3007_length__pos__if__in__set,axiom,
! [Xb: nat,Xs: list_nat] :
( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3008_length__pos__if__in__set,axiom,
! [Xb: int,Xs: list_int] :
( aa_set_int_o(member_int(Xb),set_int2(Xs))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).
% length_pos_if_in_set
tff(fact_3009_nat__diff__split,axiom,
! [P: fun_nat_o,A: nat,B: nat] :
( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
<=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(P,zero_zero_nat) )
& ! [D3: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
=> aa_nat_o(P,D3) ) ) ) ).
% nat_diff_split
tff(fact_3010_nat__diff__split__asm,axiom,
! [P: fun_nat_o,A: nat,B: nat] :
( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
<=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
& ~ aa_nat_o(P,zero_zero_nat) )
| ? [D3: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D3) )
& ~ aa_nat_o(P,D3) ) ) ) ).
% nat_diff_split_asm
tff(fact_3011_nat__mult__le__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% nat_mult_le_cancel1
tff(fact_3012_power__gt__expt,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).
% power_gt_expt
tff(fact_3013_div__le__mono2,axiom,
! [Mb: nat,Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).
% div_le_mono2
tff(fact_3014_div__greater__zero__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% div_greater_zero_iff
tff(fact_3015_div__eq__dividend__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
<=> ( Nb = one_one_nat ) ) ) ).
% div_eq_dividend_iff
tff(fact_3016_div__less__dividend,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).
% div_less_dividend
tff(fact_3017_nat__one__le__power,axiom,
! [I: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb)) ) ).
% nat_one_le_power
tff(fact_3018_div__less__iff__less__mult,axiom,
! [Q3: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q3)),Nb)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).
% div_less_iff_less_mult
tff(fact_3019_nat__mult__div__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).
% nat_mult_div_cancel1
tff(fact_3020_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2) ).
% vebt_insert.simps(3)
tff(fact_3021_vebt__succ_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).
% vebt_succ.simps(5)
tff(fact_3022_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).
% mult_less_cancel_right2
tff(fact_3023_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).
% mult_less_cancel_right2
tff(fact_3024_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).
% mult_less_cancel_right2
tff(fact_3025_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).
% mult_less_cancel_right1
tff(fact_3026_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).
% mult_less_cancel_right1
tff(fact_3027_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).
% mult_less_cancel_right1
tff(fact_3028_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).
% mult_less_cancel_left2
tff(fact_3029_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).
% mult_less_cancel_left2
tff(fact_3030_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).
% mult_less_cancel_left2
tff(fact_3031_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).
% mult_less_cancel_left1
tff(fact_3032_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).
% mult_less_cancel_left1
tff(fact_3033_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).
% mult_less_cancel_left1
tff(fact_3034_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).
% mult_le_cancel_right2
tff(fact_3035_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).
% mult_le_cancel_right2
tff(fact_3036_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).
% mult_le_cancel_right2
tff(fact_3037_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).
% mult_le_cancel_right1
tff(fact_3038_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).
% mult_le_cancel_right1
tff(fact_3039_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).
% mult_le_cancel_right1
tff(fact_3040_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).
% mult_le_cancel_left2
tff(fact_3041_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).
% mult_le_cancel_left2
tff(fact_3042_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).
% mult_le_cancel_left2
tff(fact_3043_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).
% mult_le_cancel_left1
tff(fact_3044_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).
% mult_le_cancel_left1
tff(fact_3045_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).
% mult_le_cancel_left1
tff(fact_3046_field__le__mult__one__interval,axiom,
! [Xb: rat,Y: rat] :
( ! [Z2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Xb)),Y) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).
% field_le_mult_one_interval
tff(fact_3047_field__le__mult__one__interval,axiom,
! [Xb: real,Y: real] :
( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Xb)),Y) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).
% field_le_mult_one_interval
tff(fact_3048_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).
% divide_le_eq
tff(fact_3049_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).
% divide_le_eq
tff(fact_3050_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).
% le_divide_eq
tff(fact_3051_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).
% le_divide_eq
tff(fact_3052_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).
% divide_left_mono
tff(fact_3053_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
=> aa_real_o(aa_real_fun_real_o(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_3054_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).
% neg_divide_le_eq
tff(fact_3055_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).
% neg_divide_le_eq
tff(fact_3056_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% neg_le_divide_eq
tff(fact_3057_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).
% neg_le_divide_eq
tff(fact_3058_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% pos_divide_le_eq
tff(fact_3059_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).
% pos_divide_le_eq
tff(fact_3060_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).
% pos_le_divide_eq
tff(fact_3061_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).
% pos_le_divide_eq
tff(fact_3062_mult__imp__div__pos__le,axiom,
! [Y: rat,Xb: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Z) ) ) ).
% mult_imp_div_pos_le
tff(fact_3063_mult__imp__div__pos__le,axiom,
! [Y: real,Xb: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).
% mult_imp_div_pos_le
tff(fact_3064_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% mult_imp_le_div_pos
tff(fact_3065_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% mult_imp_le_div_pos
tff(fact_3066_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).
% divide_left_mono_neg
tff(fact_3067_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
=> aa_real_o(aa_real_fun_real_o(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_3068_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).
% le_divide_eq_1
tff(fact_3069_le__divide__eq__1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).
% le_divide_eq_1
tff(fact_3070_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
<=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
tff(fact_3071_divide__le__eq__1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
<=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
| ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
tff(fact_3072_convex__bound__le,axiom,
! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
=> ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3073_convex__bound__le,axiom,
! [Xb: int,A: int,Y: int,U: int,V: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> aa_int_o(aa_int_fun_int_o(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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3074_convex__bound__le,axiom,
! [Xb: real,A: real,Y: real,U: real,V: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3075_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_3076_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_3077_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_3078_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_3079_frac__le__eq,axiom,
! [Y: rat,Z: rat,Xb: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),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_3080_frac__le__eq,axiom,
! [Y: real,Z: real,Xb: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
<=> aa_real_o(aa_real_fun_real_o(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,Xb),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_3081_power__Suc__less,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
=> aa_real_o(aa_real_fun_real_o(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),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).
% power_Suc_less
tff(fact_3082_power__Suc__less,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).
% power_Suc_less
tff(fact_3083_power__Suc__less,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(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),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).
% power_Suc_less
tff(fact_3084_power__Suc__less,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).
% power_Suc_less
tff(fact_3085_frac__less__eq,axiom,
! [Y: real,Z: real,Xb: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
<=> aa_real_o(aa_real_fun_real_o(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,Xb),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_3086_frac__less__eq,axiom,
! [Y: rat,Z: rat,Xb: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),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_3087_power__Suc__le__self,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).
% power_Suc_le_self
tff(fact_3088_power__Suc__le__self,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).
% power_Suc_le_self
tff(fact_3089_power__Suc__le__self,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),A) ) ) ).
% power_Suc_le_self
tff(fact_3090_power__Suc__le__self,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).
% power_Suc_le_self
tff(fact_3091_power__Suc__less__one,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).
% power_Suc_less_one
tff(fact_3092_power__Suc__less__one,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).
% power_Suc_less_one
tff(fact_3093_power__Suc__less__one,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).
% power_Suc_less_one
tff(fact_3094_power__Suc__less__one,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).
% power_Suc_less_one
tff(fact_3095_power__strict__decreasing,axiom,
! [Nb: nat,N4: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).
% power_strict_decreasing
tff(fact_3096_power__strict__decreasing,axiom,
! [Nb: nat,N4: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).
% power_strict_decreasing
tff(fact_3097_power__strict__decreasing,axiom,
! [Nb: nat,N4: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).
% power_strict_decreasing
tff(fact_3098_power__strict__decreasing,axiom,
! [Nb: nat,N4: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).
% power_strict_decreasing
tff(fact_3099_power__decreasing,axiom,
! [Nb: nat,N4: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).
% power_decreasing
tff(fact_3100_power__decreasing,axiom,
! [Nb: nat,N4: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).
% power_decreasing
tff(fact_3101_power__decreasing,axiom,
! [Nb: nat,N4: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).
% power_decreasing
tff(fact_3102_power__decreasing,axiom,
! [Nb: nat,N4: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).
% power_decreasing
tff(fact_3103_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_3104_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_3105_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_3106_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_3107_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_3108_self__le__power,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).
% self_le_power
tff(fact_3109_self__le__power,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).
% self_le_power
tff(fact_3110_self__le__power,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).
% self_le_power
tff(fact_3111_self__le__power,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).
% self_le_power
tff(fact_3112_one__less__power,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).
% one_less_power
tff(fact_3113_one__less__power,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).
% one_less_power
tff(fact_3114_one__less__power,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).
% one_less_power
tff(fact_3115_one__less__power,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).
% one_less_power
tff(fact_3116_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_3117_pos2,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% pos2
tff(fact_3118_power__diff,axiom,
! [A: complex,Nb: nat,Mb: nat] :
( ( A != zero_zero_complex )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).
% power_diff
tff(fact_3119_power__diff,axiom,
! [A: real,Nb: nat,Mb: nat] :
( ( A != zero_zero_real )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).
% power_diff
tff(fact_3120_power__diff,axiom,
! [A: rat,Nb: nat,Mb: nat] :
( ( A != zero_zero_rat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).
% power_diff
tff(fact_3121_power__diff,axiom,
! [A: nat,Nb: nat,Mb: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).
% power_diff
tff(fact_3122_power__diff,axiom,
! [A: int,Nb: nat,Mb: nat] :
( ( A != zero_zero_int )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).
% power_diff
tff(fact_3123_Suc__diff__eq__diff__pred,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).
% Suc_diff_eq_diff_pred
tff(fact_3124_Suc__pred_H,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).
% Suc_pred'
tff(fact_3125_add__eq__if,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,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,Mb),one_one_nat)),Nb))) ).
% add_eq_if
tff(fact_3126_div__if,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| ( Nb = zero_zero_nat ) ),
zero_zero_nat,
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,Mb),Nb)),Nb)) ) ).
% div_if
tff(fact_3127_div__geq,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = 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,Mb),Nb)),Nb)) ) ) ) ).
% div_geq
tff(fact_3128_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q3))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),Nb) ) ) ).
% less_eq_div_iff_mult_less_eq
tff(fact_3129_dividend__less__times__div,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).
% dividend_less_times_div
tff(fact_3130_dividend__less__div__times,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).
% dividend_less_div_times
tff(fact_3131_split__div,axiom,
! [P: fun_nat_o,Mb: nat,Nb: nat] :
( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
<=> ( ( ( Nb = zero_zero_nat )
=> aa_nat_o(P,zero_zero_nat) )
& ( ( Nb != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
=> ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I4)),J3) )
=> aa_nat_o(P,I4) ) ) ) ) ) ).
% split_div
tff(fact_3132_mult__eq__if,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).
% mult_eq_if
tff(fact_3133_mult__commute__abs,axiom,
! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_as,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).
% mult_commute_abs
tff(fact_3134_mult__commute__abs,axiom,
! [C: rat] : aTP_Lamm_at(C) = times_times_rat(C) ).
% mult_commute_abs
tff(fact_3135_mult__commute__abs,axiom,
! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_au,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).
% mult_commute_abs
tff(fact_3136_mult__commute__abs,axiom,
! [C: int] : aTP_Lamm_av(C) = aa_int_fun_int_int(times_times_int,C) ).
% mult_commute_abs
tff(fact_3137_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz)),Va) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_3138_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).
% vebt_pred.simps(5)
tff(fact_3139_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve)),Vf) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_3140_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,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_3141_convex__bound__lt,axiom,
! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
=> ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_3142_convex__bound__lt,axiom,
! [Xb: int,A: int,Y: int,U: int,V: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> aa_int_o(aa_int_fun_int_o(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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_3143_convex__bound__lt,axiom,
! [Xb: real,A: real,Y: real,U: real,V: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_3144_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_3145_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_3146_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_3147_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_3148_half__gt__zero,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).
% half_gt_zero
tff(fact_3149_half__gt__zero,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).
% half_gt_zero
tff(fact_3150_half__gt__zero__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).
% half_gt_zero_iff
tff(fact_3151_half__gt__zero__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).
% half_gt_zero_iff
tff(fact_3152_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S2))),V) ) ) ) ).
% scaling_mono
tff(fact_3153_scaling__mono,axiom,
! [U: real,V: real,R2: real,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S2)
=> aa_real_o(aa_real_fun_real_o(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,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S2))),V) ) ) ) ).
% scaling_mono
tff(fact_3154_zero__le__power2,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(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_3155_zero__le__power2,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(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_3156_zero__le__power2,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(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_3157_power2__eq__imp__eq,axiom,
! [Xb: rat,Y: rat] :
( ( aa_nat_rat(power_power_rat(Xb),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_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( Xb = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3158_power2__eq__imp__eq,axiom,
! [Xb: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
=> ( Xb = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3159_power2__eq__imp__eq,axiom,
! [Xb: int,Y: int] :
( ( aa_nat_int(power_power_int(Xb),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_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( Xb = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3160_power2__eq__imp__eq,axiom,
! [Xb: real,Y: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( Xb = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3161_power2__le__imp__le,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),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_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).
% power2_le_imp_le
tff(fact_3162_power2__le__imp__le,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).
% power2_le_imp_le
tff(fact_3163_power2__le__imp__le,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),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_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).
% power2_le_imp_le
tff(fact_3164_power2__le__imp__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).
% power2_le_imp_le
tff(fact_3165_power2__less__0,axiom,
! [A: real] : ~ aa_real_o(aa_real_fun_real_o(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_3166_power2__less__0,axiom,
! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(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_3167_power2__less__0,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(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_3168_exp__add__not__zero__imp__left,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)) != 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))),Mb) != zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
tff(fact_3169_exp__add__not__zero__imp__left,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)) != zero_zero_int )
=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
tff(fact_3170_exp__add__not__zero__imp__right,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)) != 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))),Nb) != zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
tff(fact_3171_exp__add__not__zero__imp__right,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)) != zero_zero_int )
=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
tff(fact_3172_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [Nb: nat,Mb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != 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,Nb),Mb)) != zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3173_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [Nb: nat,Mb: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != 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,Nb),Mb)) != zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3174_power__diff__power__eq,axiom,
! [A: nat,Mb: nat,Nb: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),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,Nb),Mb)))) ) ) ).
% power_diff_power_eq
tff(fact_3175_power__diff__power__eq,axiom,
! [A: int,Mb: nat,Nb: nat] :
( ( A != zero_zero_int )
=> ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_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,Nb),Mb)))) ) ) ).
% power_diff_power_eq
tff(fact_3176_less__2__cases,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
=> ( ( Nb = zero_zero_nat )
| ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% less_2_cases
tff(fact_3177_less__2__cases__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
<=> ( ( Nb = zero_zero_nat )
| ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% less_2_cases_iff
tff(fact_3178_nat__induct2,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,zero_zero_nat)
=> ( aa_nat_o(P,one_one_nat)
=> ( ! [N: nat] :
( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
=> aa_nat_o(P,Nb) ) ) ) ).
% nat_induct2
tff(fact_3179_power__eq__if,axiom,
! [P2: complex,Mb: nat] :
aa_nat_complex(power_power_complex(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,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,Mb),one_one_nat)))) ).
% power_eq_if
tff(fact_3180_power__eq__if,axiom,
! [P2: real,Mb: nat] :
aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,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,Mb),one_one_nat)))) ).
% power_eq_if
tff(fact_3181_power__eq__if,axiom,
! [P2: rat,Mb: nat] :
aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,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,Mb),one_one_nat)))) ).
% power_eq_if
tff(fact_3182_power__eq__if,axiom,
! [P2: nat,Mb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,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,Mb),one_one_nat)))) ).
% power_eq_if
tff(fact_3183_power__eq__if,axiom,
! [P2: int,Mb: nat] :
aa_nat_int(power_power_int(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_int,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,Mb),one_one_nat)))) ).
% power_eq_if
tff(fact_3184_power__minus__mult,axiom,
! [Nb: nat,A: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).
% power_minus_mult
tff(fact_3185_power__minus__mult,axiom,
! [Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( 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,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).
% power_minus_mult
tff(fact_3186_power__minus__mult,axiom,
! [Nb: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).
% power_minus_mult
tff(fact_3187_power__minus__mult,axiom,
! [Nb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( 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,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).
% power_minus_mult
tff(fact_3188_power__minus__mult,axiom,
! [Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( 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,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).
% power_minus_mult
tff(fact_3189_le__div__geq,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = 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,Mb),Nb)),Nb)) ) ) ) ).
% le_div_geq
tff(fact_3190_split__div_H,axiom,
! [P: fun_nat_o,Mb: nat,Nb: nat] :
( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
<=> ( ( ( Nb = zero_zero_nat )
& aa_nat_o(P,zero_zero_nat) )
| ? [Q4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Mb)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
& aa_nat_o(P,Q4) ) ) ) ).
% split_div'
tff(fact_3191_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).
% vebt_pred.simps(6)
tff(fact_3192_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)),Vj) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_3193_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,Xb: 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)),Xb) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_3194_power2__less__imp__less,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),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_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).
% power2_less_imp_less
tff(fact_3195_power2__less__imp__less,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).
% power2_less_imp_less
tff(fact_3196_power2__less__imp__less,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),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_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).
% power2_less_imp_less
tff(fact_3197_power2__less__imp__less,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).
% power2_less_imp_less
tff(fact_3198_sum__power2__le__zero__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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)
<=> ( ( Xb = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_3199_sum__power2__le__zero__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),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)
<=> ( ( Xb = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_3200_sum__power2__le__zero__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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)
<=> ( ( Xb = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_3201_sum__power2__ge__zero,axiom,
! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_3202_sum__power2__ge__zero,axiom,
! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),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_3203_sum__power2__ge__zero,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_3204_not__sum__power2__lt__zero,axiom,
! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_3205_not__sum__power2__lt__zero,axiom,
! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_3206_not__sum__power2__lt__zero,axiom,
! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),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_3207_sum__power2__gt__zero__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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)))))
<=> ( ( Xb != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_3208_sum__power2__gt__zero__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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)))))
<=> ( ( Xb != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_3209_sum__power2__gt__zero__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),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)))))
<=> ( ( Xb != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_3210_zero__le__even__power_H,axiom,
! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(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))),Nb))) ).
% zero_le_even_power'
tff(fact_3211_zero__le__even__power_H,axiom,
! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(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))),Nb))) ).
% zero_le_even_power'
tff(fact_3212_zero__le__even__power_H,axiom,
! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(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))),Nb))) ).
% zero_le_even_power'
tff(fact_3213_nat__bit__induct,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,zero_zero_nat)
=> ( ! [N: nat] :
( aa_nat_o(P,N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
=> ( ! [N: nat] :
( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
=> aa_nat_o(P,Nb) ) ) ) ).
% nat_bit_induct
tff(fact_3214_div__2__gt__zero,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).
% div_2_gt_zero
tff(fact_3215_Suc__n__div__2__gt__zero,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).
% Suc_n_div_2_gt_zero
tff(fact_3216_odd__0__le__power__imp__0__le,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(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))),Nb))))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% odd_0_le_power_imp_0_le
tff(fact_3217_odd__0__le__power__imp__0__le,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(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))),Nb))))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% odd_0_le_power_imp_0_le
tff(fact_3218_odd__0__le__power__imp__0__le,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(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))),Nb))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% odd_0_le_power_imp_0_le
tff(fact_3219_odd__power__less__zero,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(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))),Nb)))),zero_zero_real) ) ).
% odd_power_less_zero
tff(fact_3220_odd__power__less__zero,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(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))),Nb)))),zero_zero_rat) ) ).
% odd_power_less_zero
tff(fact_3221_odd__power__less__zero,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(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))),Nb)))),zero_zero_int) ) ).
% odd_power_less_zero
tff(fact_3222_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [Xb: nat,Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Nb),Mb)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
tff(fact_3223_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [Xb: nat,Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Nb),Mb)))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
tff(fact_3224_succ_H__bound__height,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).
% succ'_bound_height
tff(fact_3225_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
=> ( ! [Mi: nat,Ma: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
=> ~ ( ( Xaa = Mi )
| ( Xaa = Ma ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
=> ~ ( ( Xaa = Mi )
| ( Xaa = Ma )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
=> ~ $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
tff(fact_3226_arith__geo__mean,axiom,
! [U: rat,Xb: 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(Xb),Y) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).
% arith_geo_mean
tff(fact_3227_arith__geo__mean,axiom,
! [U: real,Xb: 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,Xb),Y) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).
% arith_geo_mean
tff(fact_3228_vebt__member_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),Xb) ).
% vebt_member.simps(2)
tff(fact_3229_is__succ__in__set__def,axiom,
! [Xs: set_nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y)
<=> ( aa_set_nat_o(member_nat(Y),Xs)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
& ! [X: nat] :
( aa_set_nat_o(member_nat(X),Xs)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ) ) ).
% is_succ_in_set_def
tff(fact_3230_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_3231_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd)),Xb)
<=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).
% VEBT_internal.membermima.simps(5)
tff(fact_3232_vebt__succ_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
( ( vEBT_vebt_succ(Xb,Xaa) = Y )
=> ( ! [Uu: $o,B4: $o] :
( ( Xb = vEBT_Leaf((Uu),(B4)) )
=> ( ( Xaa = zero_zero_nat )
=> ( Y != $ite((B4),some_nat(one_one_nat),none_nat) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
=> ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
=> ( Y != none_nat ) ) )
=> ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
some_nat(Mi),
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summary,h),
$ite(sc = 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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
tff(fact_3233_vebt__pred_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
( ( vEBT_vebt_pred(Xb,Xaa) = Y )
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> ( ( Xaa = zero_zero_nat )
=> ( Y != none_nat ) ) )
=> ( ! [A4: $o] :
( ? [Uw: $o] : Xb = vEBT_Leaf((A4),(Uw))
=> ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
=> ( Y != $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
=> ( Y != $ite(
(B4),
some_nat(one_one_nat),
$ite((A4),some_nat(zero_zero_nat),none_nat) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
some_nat(Ma),
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summary,h),
$ite(
pr = none_nat,
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
tff(fact_3234_buildup__gives__valid,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).
% buildup_gives_valid
tff(fact_3235_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xb),Xaa) = Y )
=> ( ( ? [Uu: $o,B4: $o] : Xb = vEBT_Leaf((Uu),(B4))
=> ( ( Xaa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
=> ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
=> ( Y != one_one_nat ) ) )
=> ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList,h)),l)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary),h)),one_one_nat) ) ),
one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_3236_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xb),Xaa) = Y )
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> ( ( Xaa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ? [A4: $o,Uw: $o] : Xb = vEBT_Leaf((A4),(Uw))
=> ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
=> ( Y != one_one_nat ) ) )
=> ( ( ? [A4: $o,B4: $o] : Xb = vEBT_Leaf((A4),(B4))
=> ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
=> ( Y != one_one_nat ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList,h)),l)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary),h)),one_one_nat) ) ),
one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_3237_inrange,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),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))),Nb)),one_one_nat))) ) ).
% inrange
tff(fact_3238_buildup__gives__empty,axiom,
! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).
% buildup_gives_empty
tff(fact_3239_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_3240_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_3241_buildup__nothing__in__min__max,axiom,
! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).
% buildup_nothing_in_min_max
tff(fact_3242_buildup__nothing__in__leaf,axiom,
! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).
% buildup_nothing_in_leaf
tff(fact_3243_Leaf__0__not,axiom,
! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).
% Leaf_0_not
tff(fact_3244_deg1Leaf,axiom,
! [Ta: vEBT_VEBT] :
( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
<=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).
% deg1Leaf
tff(fact_3245_deg__1__Leaf,axiom,
! [Ta: vEBT_VEBT] :
( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
=> ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ).
% deg_1_Leaf
tff(fact_3246_deg__1__Leafy,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( Nb = one_one_nat )
=> ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ) ).
% deg_1_Leafy
tff(fact_3247_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y22)) )
<=> ( ( (X21)
<=> (Y21) )
& ( (X222)
<=> (Y22) ) ) ) ).
% VEBT.inject(2)
tff(fact_3248_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
=> ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).
% div_pos_pos_trivial
tff(fact_3249_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
=> ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).
% div_neg_neg_trivial
tff(fact_3250_i0__less,axiom,
! [Nb: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
<=> ( Nb != zero_z5237406670263579293d_enat ) ) ).
% i0_less
tff(fact_3251_idiff__0,axiom,
! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).
% idiff_0
tff(fact_3252_idiff__0__right,axiom,
! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).
% idiff_0_right
tff(fact_3253_not__real__square__gt__zero,axiom,
! [Xb: real] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb))
<=> ( Xb = zero_zero_real ) ) ).
% not_real_square_gt_zero
tff(fact_3254_half__negative__int__iff,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% half_negative_int_iff
tff(fact_3255_half__nonnegative__int__iff,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).
% half_nonnegative_int_iff
tff(fact_3256_vebt__buildup_Osimps_I1_J,axiom,
vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).
% vebt_buildup.simps(1)
tff(fact_3257_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).
% VEBT.size(4)
tff(fact_3258_vebt__buildup_Osimps_I2_J,axiom,
vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).
% vebt_buildup.simps(2)
tff(fact_3259_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
tff(fact_3260_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).
% VEBT.distinct(1)
tff(fact_3261_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: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).
% VEBT.exhaust
tff(fact_3262_VEBT__internal_Ovalid_H_Ocases,axiom,
! [Xb: produc9072475918466114483BT_nat] :
( ! [Uu: $o,Uv: $o,D4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),D4)
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).
% VEBT_internal.valid'.cases
tff(fact_3263_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,Uw2: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu2),(Uv2))),Uw2) ).
% VEBT_internal.membermima.simps(1)
tff(fact_3264_int__div__less__self,axiom,
! [Xb: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(Xb),K)),Xb) ) ) ).
% int_div_less_self
tff(fact_3265_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).
% pos_imp_zdiv_neg_iff
tff(fact_3266_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).
% neg_imp_zdiv_neg_iff
tff(fact_3267_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).
% div_neg_pos_less0
tff(fact_3268_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).
% zdiv_zmult2_eq
tff(fact_3269_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
tff(fact_3270_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).
% pos_imp_zdiv_nonneg_iff
tff(fact_3271_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).
% neg_imp_zdiv_nonneg_iff
tff(fact_3272_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(I),K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).
% pos_imp_zdiv_pos_iff
tff(fact_3273_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).
% div_nonpos_pos_le0
tff(fact_3274_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).
% div_nonneg_neg_le0
tff(fact_3275_div__positive__int,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K),L)) ) ) ).
% div_positive_int
tff(fact_3276_div__int__pos__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),L))
<=> ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).
% div_int_pos_iff
tff(fact_3277_zdiv__mono2__neg,axiom,
! [A: int,B2: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B2)),aa_int_int(divide_divide_int(A),B)) ) ) ) ).
% zdiv_mono2_neg
tff(fact_3278_zdiv__mono1__neg,axiom,
! [A: int,A2: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A2),B)),aa_int_int(divide_divide_int(A),B)) ) ) ).
% zdiv_mono1_neg
tff(fact_3279_int__div__pos__eq,axiom,
! [A: int,B: int,Q3: int,R2: 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),Q3)),R2) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
=> ( aa_int_int(divide_divide_int(A),B) = Q3 ) ) ) ) ).
% int_div_pos_eq
tff(fact_3280_int__div__neg__eq,axiom,
! [A: int,B: int,Q3: int,R2: 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),Q3)),R2) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
=> ( aa_int_int(divide_divide_int(A),B) = Q3 ) ) ) ) ).
% int_div_neg_eq
tff(fact_3281_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( aa_int_int(divide_divide_int(I),K) = zero_zero_int )
<=> ( ( K = zero_zero_int )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
& aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).
% zdiv_eq_0_iff
tff(fact_3282_div__pos__geq,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
=> ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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_3283_zdiv__mono2,axiom,
! [A: int,B2: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B2)) ) ) ) ).
% zdiv_mono2
tff(fact_3284_zdiv__mono1,axiom,
! [A: int,A2: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A2),B)) ) ) ).
% zdiv_mono1
tff(fact_3285_split__zdiv,axiom,
! [P: fun_int_o,Nb: int,K: int] :
( aa_int_o(P,aa_int_int(divide_divide_int(Nb),K))
<=> ( ( ( K = zero_zero_int )
=> aa_int_o(P,zero_zero_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ! [I4: int,J3: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
& ( Nb = 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) ) )
=> aa_int_o(P,I4) ) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
=> ! [I4: int,J3: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
& ( Nb = 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) ) )
=> aa_int_o(P,I4) ) ) ) ) ).
% split_zdiv
tff(fact_3286_enat__0__less__mult__iff,axiom,
! [Mb: extended_enat,Nb: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(Mb,Nb))
<=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Mb)
& aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb) ) ) ).
% enat_0_less_mult_iff
tff(fact_3287_not__iless0,axiom,
! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),zero_z5237406670263579293d_enat) ).
% not_iless0
tff(fact_3288_iadd__is__0,axiom,
! [Mb: extended_enat,Nb: extended_enat] :
( ( plus_p3455044024723400733d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
<=> ( ( Mb = zero_z5237406670263579293d_enat )
& ( Nb = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
tff(fact_3289_ile0__eq,axiom,
! [Nb: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),zero_z5237406670263579293d_enat)
<=> ( Nb = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
tff(fact_3290_i0__lb,axiom,
! [Nb: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Nb) ).
% i0_lb
tff(fact_3291_ex__nat__less,axiom,
! [Nb: nat,P: fun_nat_o] :
( ? [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
& aa_nat_o(P,M4) )
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
& aa_nat_o(P,X) ) ) ).
% ex_nat_less
tff(fact_3292_all__nat__less,axiom,
! [Nb: nat,P: fun_nat_o] :
( ! [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
=> aa_nat_o(P,M4) )
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
=> aa_nat_o(P,X) ) ) ).
% all_nat_less
tff(fact_3293_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: $o,B: $o,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A),(B))),Xb) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_3294_VEBT__internal_Onaive__member_Ocases,axiom,
! [Xb: produc9072475918466114483BT_nat] :
( ! [A4: $o,B4: $o,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X4)
=> ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Ux2)
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),X4) ) ) ).
% VEBT_internal.naive_member.cases
tff(fact_3295_invar__vebt_Ointros_I1_J,axiom,
! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).
% invar_vebt.intros(1)
tff(fact_3296_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
! [Xb: vEBT_VEBT] :
( ( Xb != vEBT_Leaf($false,$false) )
=> ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
=> ( ! [Uu: $o] : Xb != vEBT_Leaf((Uu),$true)
=> ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_3297_vebt__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,Xb: nat] :
( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),Xb)
<=> $ite(
Xb = zero_zero_nat,
(A),
$ite(Xb = one_one_nat,(B),$false) ) ) ).
% vebt_member.simps(1)
tff(fact_3298_vebt__insert_Osimps_I1_J,axiom,
! [A: $o,B: $o,Xb: nat] :
vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
Xb = zero_zero_nat,
vEBT_Leaf($true,(B)),
$ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).
% vebt_insert.simps(1)
tff(fact_3299_vebt__succ_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,Nb)) = none_nat ).
% vebt_succ.simps(2)
tff(fact_3300_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,Xb: nat] :
( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),Xb)
<=> $ite(
Xb = zero_zero_nat,
(A),
$ite(Xb = one_one_nat,(B),$false) ) ) ).
% VEBT_internal.naive_member.simps(1)
tff(fact_3301_vebt__pred_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat) = none_nat ).
% vebt_pred.simps(1)
tff(fact_3302_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
! [A: $o,B: $o,Va: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_3303_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_3304_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,Nb: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,Nb)) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_3305_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
! [Uu2: $o,B: $o] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Leaf((Uu2),(B))),zero_zero_nat) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_3306_not__exp__less__eq__0__int,axiom,
! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).
% not_exp_less_eq_0_int
tff(fact_3307_realpow__pos__nth2,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ? [R3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).
% realpow_pos_nth2
tff(fact_3308_real__arch__pow__inv,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),Y) ) ) ).
% real_arch_pow_inv
tff(fact_3309_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [Xb: vEBT_VEBT] :
( ! [A4: $o,B4: $o] : Xb != vEBT_Leaf((A4),(B4))
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_3310_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
! [A: $o,Uw2: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((A),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_3311_realpow__pos__nth,axiom,
! [Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ? [R3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = A ) ) ) ) ).
% realpow_pos_nth
tff(fact_3312_realpow__pos__nth__unique,axiom,
! [Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),Nb) = A )
& ! [Y4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),Nb) = A ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
tff(fact_3313_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_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(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_3314_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_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(divide_divide_int(B),A) ) ) ).
% pos_zdiv_mult_2
tff(fact_3315_vebt__pred_Osimps_I2_J,axiom,
! [A: $o,Uw2: $o] :
vEBT_vebt_pred(vEBT_Leaf((A),(Uw2)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),some_nat(zero_zero_nat),none_nat) ).
% vebt_pred.simps(2)
tff(fact_3316_vebt__mint_Osimps_I1_J,axiom,
! [A: $o,B: $o] :
vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
(A),
some_nat(zero_zero_nat),
$ite((B),some_nat(one_one_nat),none_nat) ) ).
% vebt_mint.simps(1)
tff(fact_3317_vebt__succ_Osimps_I1_J,axiom,
! [Uu2: $o,B: $o] :
vEBT_vebt_succ(vEBT_Leaf((Uu2),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).
% vebt_succ.simps(1)
tff(fact_3318_vebt__maxt_Osimps_I1_J,axiom,
! [A: $o,B: $o] :
vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
(B),
some_nat(one_one_nat),
$ite((A),some_nat(zero_zero_nat),none_nat) ) ).
% vebt_maxt.simps(1)
tff(fact_3319_int__power__div__base,axiom,
! [Mb: nat,K: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),Mb)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).
% int_power_div_base
tff(fact_3320_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [Xb: produc9072475918466114483BT_nat] :
( ! [Uu: $o,Uv: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)
=> ( ! [A4: $o,Uw: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw)),aa_nat_nat(suc,zero_zero_nat))
=> ( ! [A4: $o,B4: $o,Va2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2)
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X4) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_3321_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [Xb: produc9072475918466114483BT_nat] :
( ! [Uu: $o,B4: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B4)),zero_zero_nat)
=> ( ! [Uv: $o,Uw: $o,N: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))
=> ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
=> ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve2)
=> ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X4) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_3322_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,
! [Xb: produc9072475918466114483BT_nat] :
( ! [A4: $o,B4: $o,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X4)
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),X4)
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X4)
=> ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),X4)
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X4) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_3323_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,
! [Xb: produc9072475918466114483BT_nat] :
( ! [A4: $o,B4: $o,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X4)
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),X4)
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X4)
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X4)
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X4) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_3324_VEBT__internal_Omembermima_Ocases,axiom,
! [Xb: produc9072475918466114483BT_nat] :
( ! [Uu: $o,Uv: $o,Uw: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Uw)
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X4)
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X4)
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),X4) ) ) ) ) ).
% VEBT_internal.membermima.cases
tff(fact_3325_vebt__pred_Osimps_I3_J,axiom,
! [A: $o,B: $o,Va: nat] :
vEBT_vebt_pred(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
(B),
some_nat(one_one_nat),
$ite((A),some_nat(zero_zero_nat),none_nat) ) ).
% vebt_pred.simps(3)
tff(fact_3326_vebt__mint_Oelims,axiom,
! [Xb: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_mint(Xb) = Y )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( Y != $ite(
(A4),
some_nat(zero_zero_nat),
$ite((B4),some_nat(one_one_nat),none_nat) ) ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat] :
( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
=> ( Y != some_nat(Mi) ) ) ) ) ) ).
% vebt_mint.elims
tff(fact_3327_vebt__maxt_Oelims,axiom,
! [Xb: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_maxt(Xb) = Y )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( Y != $ite(
(B4),
some_nat(one_one_nat),
$ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( Y != none_nat ) )
=> ~ ! [Mi: nat,Ma: nat] :
( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
=> ( Y != some_nat(Ma) ) ) ) ) ) ).
% vebt_maxt.elims
tff(fact_3328_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
<=> (Y) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( (Y)
<=> ~ $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) ) )
=> ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
=> (Y) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
=> ( (Y)
<=> ~ $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
tff(fact_3329_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ~ $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
=> ~ $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
tff(fact_3330_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) )
=> ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
tff(fact_3331_vebt__member_Oelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ~ $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
=> ~ $ite(
Xaa = Mi,
$true,
$ite(
Xaa = Ma,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
tff(fact_3332_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
=> ( ! [Uu: $o,Uv: $o] : Xb != vEBT_Leaf((Uu),(Uv))
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
=> ( ! [Mi: nat,Ma: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
=> ( ( Xaa = Mi )
| ( Xaa = Ma ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
=> ( ( Xaa = Mi )
| ( Xaa = Ma )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
tff(fact_3333_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
<=> (Y) )
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> (Y) )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
=> (Y) )
=> ( ! [Mi: nat,Ma: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
=> ( (Y)
<=> ~ ( ( Xaa = Mi )
| ( Xaa = Ma ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
=> ( (Y)
<=> ~ ( ( Xaa = Mi )
| ( Xaa = Ma )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
=> ( (Y)
<=> ~ $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
tff(fact_3334_vebt__member_Oelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
=> $ite(
Xaa = Mi,
$true,
$ite(
Xaa = Ma,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
tff(fact_3335_vebt__member_Oelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
<=> (Y) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( (Y)
<=> ~ $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> (Y) )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
=> (Y) )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> (Y) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
=> ( (Y)
<=> ~ $ite(
Xaa = Mi,
$true,
$ite(
Xaa = Ma,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
tff(fact_3336_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xb),Xaa) = Y )
=> ( ( ? [A4: $o,B4: $o] : Xb = vEBT_Leaf((A4),(B4))
=> ( Y != one_one_nat ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = 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] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
Xaa = Mi,
zero_zero_nat,
$ite(
Xaa = Ma,
zero_zero_nat,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
zero_zero_nat,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
zero_zero_nat,
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma) ),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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_3337_invar__vebt_Osimps,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( aa_nat_o(vEBT_invar_vebt(A1),A22)
<=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
& ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
| ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> aa_nat_o(vEBT_invar_vebt(X),N3) )
& aa_nat_o(vEBT_invar_vebt(Summary2),N3)
& ( 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))),N3) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
& ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
| ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> aa_nat_o(vEBT_invar_vebt(X),N3) )
& aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3))
& ( 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(suc,N3)) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
& ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
| ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> aa_nat_o(vEBT_invar_vebt(X),N3) )
& aa_nat_o(vEBT_invar_vebt(Summary2),N3)
& ( 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))),N3) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),N3))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I4) ) )
& ( ( Mi2 = Ma2 )
=> ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
& aa_nat_o(aa_nat_fun_nat_o(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))),A22))
& ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),N3))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
& ! [X: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N3) = I4 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N3)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) )
| ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> aa_nat_o(vEBT_invar_vebt(X),N3) )
& aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3))
& ( 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(suc,N3)) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,N3)))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I4) ) )
& ( ( Mi2 = Ma2 )
=> ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
=> ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
& aa_nat_o(aa_nat_fun_nat_o(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))),A22))
& ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,N3)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
& ! [X: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N3) = I4 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N3)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
tff(fact_3338_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( aa_nat_o(vEBT_invar_vebt(A1),A22)
=> ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
=> ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
=> ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
=> ( ( A22 = Deg )
=> ( ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_invar_vebt(X3),N) )
=> ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
=> ( ( 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))),M2) )
=> ( ( M2 = N )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
=> ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
=> ~ ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
=> ( ( A22 = Deg )
=> ( ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_invar_vebt(X3),N) )
=> ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
=> ( ( 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))),M2) )
=> ( ( M2 = aa_nat_nat(suc,N) )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
=> ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
=> ~ ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
=> ( ( A22 = Deg )
=> ( ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_invar_vebt(X3),N) )
=> ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
=> ( ( 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))),M2) )
=> ( ( M2 = N )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
=> ( ! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),M2))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I2) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
=> ( aa_nat_o(aa_nat_fun_nat_o(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] :
( aa_nat_o(aa_nat_fun_nat_o(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))),M2))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
=> aa_nat_o(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 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
=> ( ( A22 = Deg )
=> ( ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_invar_vebt(X3),N) )
=> ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
=> ( ( 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))),M2) )
=> ( ( M2 = aa_nat_nat(suc,N) )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
=> ( ! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),M2))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I2) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
=> ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
=> ( aa_nat_o(aa_nat_fun_nat_o(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] :
( aa_nat_o(aa_nat_fun_nat_o(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))),M2))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
=> aa_nat_o(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 )
& aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
tff(fact_3339_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).
% atLeastatMost_empty
tff(fact_3340_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).
% atLeastatMost_empty
tff(fact_3341_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).
% atLeastatMost_empty
tff(fact_3342_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).
% atLeastatMost_empty
tff(fact_3343_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).
% atLeastatMost_empty
tff(fact_3344_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))
<=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
| ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
& aa_set_nat_o(ord_less_eq_set_nat(B),D) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3345_atLeastatMost__subset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
<=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3346_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
<=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
| ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3347_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3348_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
<=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3349_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
<=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3350_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
<=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_3351_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
<=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_3352_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
<=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_3353_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_3354_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_3355_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
<=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_3356_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
<=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_3357_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
<=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_3358_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
<=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_3359_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_3360_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_3361_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
<=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_3362_vebt__succ_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
( ( vEBT_vebt_succ(Xb,Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,B4: $o] :
( ( Xb = vEBT_Leaf((Uu),(B4)) )
=> ( ( Xaa = zero_zero_nat )
=> ( ( Y = $ite((B4),some_nat(one_one_nat),none_nat) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B4)),zero_zero_nat)) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( Xb = vEBT_Leaf((Uv),(Uw)) )
=> ! [N: nat] :
( ( Xaa = aa_nat_nat(suc,N) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
=> ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
=> ( ( 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),Vg2,Vh2),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
some_nat(Mi),
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summary,h),
$ite(sc = 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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
none_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
tff(fact_3363_vebt__pred_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
( ( vEBT_vebt_pred(Xb,Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( ( Xaa = zero_zero_nat )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( Xb = vEBT_Leaf((A4),(Uw)) )
=> ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
=> ( ( Y = $ite((A4),some_nat(zero_zero_nat),none_nat) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ! [Va2: nat] :
( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
=> ( ( Y = $ite(
(B4),
some_nat(one_one_nat),
$ite((A4),some_nat(zero_zero_nat),none_nat) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
=> ( ( 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),Vh2,Vi2),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
some_nat(Ma),
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summary,h),
$ite(
pr = none_nat,
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
none_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
tff(fact_3364_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 ) )
| ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
& ~ aa_set_nat_o(ord_less_eq_set_nat(L2),H2) ) ) ) ).
% Icc_eq_Icc
tff(fact_3365_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 ) )
| ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
& ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L2),H2) ) ) ) ).
% Icc_eq_Icc
tff(fact_3366_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 ) )
| ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
& ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L2),H2) ) ) ) ).
% Icc_eq_Icc
tff(fact_3367_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 ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).
% Icc_eq_Icc
tff(fact_3368_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 ) )
| ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L2),H2) ) ) ) ).
% Icc_eq_Icc
tff(fact_3369_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 ) )
| ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
& ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),H2) ) ) ) ).
% Icc_eq_Icc
tff(fact_3370_not__min__Null__member,axiom,
! [Ta: vEBT_VEBT] :
( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
=> ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_13) ) ).
% not_min_Null_member
tff(fact_3371_min__Null__member,axiom,
! [Ta: vEBT_VEBT,Xb: nat] :
( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
=> ~ aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ).
% min_Null_member
tff(fact_3372_minNullmin,axiom,
! [Ta: vEBT_VEBT] :
( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
=> ( vEBT_vebt_mint(Ta) = none_nat ) ) ).
% minNullmin
tff(fact_3373_minminNull,axiom,
! [Ta: vEBT_VEBT] :
( ( vEBT_vebt_mint(Ta) = none_nat )
=> aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).
% minminNull
tff(fact_3374_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
<=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
& aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).
% atLeastAtMost_iff
tff(fact_3375_atLeastAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( aa_set_rat_o(member_rat(I),set_or633870826150836451st_rat(L,U))
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).
% atLeastAtMost_iff
tff(fact_3376_atLeastAtMost__iff,axiom,
! [I: num,L: num,U: num] :
( aa_set_num_o(member_num(I),set_or7049704709247886629st_num(L,U))
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).
% atLeastAtMost_iff
tff(fact_3377_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).
% atLeastAtMost_iff
tff(fact_3378_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( aa_set_int_o(member_int(I),set_or1266510415728281911st_int(L,U))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).
% atLeastAtMost_iff
tff(fact_3379_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( aa_set_real_o(member_real(I),set_or1222579329274155063t_real(L,U))
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).
% atLeastAtMost_iff
tff(fact_3380_set__bit__negative__int__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(Nb),K)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% set_bit_negative_int_iff
tff(fact_3381_q__pos__lemma,axiom,
! [B2: int,Q5: int,R4: int] :
( aa_int_o(aa_int_fun_int_o(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,B2),Q5)),R4))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).
% q_pos_lemma
tff(fact_3382_zdiv__mono2__lemma,axiom,
! [B: int,Q3: int,R2: int,B2: 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),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4) )
=> ( aa_int_o(aa_int_fun_int_o(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,B2),Q5)),R4))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
tff(fact_3383_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q3: int,R2: int,B2: 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),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4) )
=> ( aa_int_o(aa_int_fun_int_o(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,B2),Q5)),R4)),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
tff(fact_3384_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(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),Q3)),R2))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ).
% unique_quotient_lemma
tff(fact_3385_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(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),Q3)),R2))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ).
% unique_quotient_lemma_neg
tff(fact_3386_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
tff(fact_3387_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
tff(fact_3388_imult__is__0,axiom,
! [Mb: extended_enat,Nb: extended_enat] :
( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
<=> ( ( Mb = zero_z5237406670263579293d_enat )
| ( Nb = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
tff(fact_3389_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) ).
% VEBT_internal.minNull.simps(5)
tff(fact_3390_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux,Uy)) ).
% VEBT_internal.minNull.simps(4)
tff(fact_3391_bounded__Max__nat,axiom,
! [P: fun_nat_o,Xb: nat,M6: nat] :
( aa_nat_o(P,Xb)
=> ( ! [X4: nat] :
( aa_nat_o(P,X4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M6) )
=> ~ ! [M2: nat] :
( aa_nat_o(P,M2)
=> ~ ! [X3: nat] :
( aa_nat_o(P,X3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M2) ) ) ) ) ).
% bounded_Max_nat
tff(fact_3392_fold__atLeastAtMost__nat_Ocases,axiom,
! [Xb: produc8114805097830739791at_num] :
~ ! [F2: fun_nat_fun_num_num,A4: nat,B4: nat,Acc: num] : Xb != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A4,product_Pair_nat_num(B4,Acc))) ).
% fold_atLeastAtMost_nat.cases
tff(fact_3393_fold__atLeastAtMost__nat_Ocases,axiom,
! [Xb: produc9217583074051984497at_nat] :
~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B4: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A4,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B4),Acc))) ).
% fold_atLeastAtMost_nat.cases
tff(fact_3394_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [Xb: vEBT_VEBT] :
( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
=> ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
=> ( ! [Uu: $o] : Xb != vEBT_Leaf((Uu),$true)
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).
% VEBT_internal.minNull.elims(3)
tff(fact_3395_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [Xb: vEBT_VEBT] :
( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
=> ( ( Xb != vEBT_Leaf($false,$false) )
=> ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) ) ) ).
% VEBT_internal.minNull.elims(2)
tff(fact_3396_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Y: $o] :
( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
<=> (Y) )
=> ( ( ( Xb = vEBT_Leaf($false,$false) )
=> ~ (Y) )
=> ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
=> (Y) )
=> ( ( ? [Uu: $o] : Xb = vEBT_Leaf((Uu),$true)
=> (Y) )
=> ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
=> ~ (Y) )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
=> (Y) ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
tff(fact_3397_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))
<=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
| ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
& aa_set_nat_o(ord_less_eq_set_nat(B),D)
& ( aa_set_nat_o(ord_less_set_nat(C),A)
| aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
& aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3398_atLeastatMost__psubset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
<=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
| ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3399_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
<=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
| ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
& ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
| aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3400_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
<=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
& ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3401_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
<=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
| aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3402_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
<=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
| ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
| aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3403_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,B4: $o] :
( ( Xb = vEBT_Leaf((Uu),(B4)) )
=> ( ( Xaa = zero_zero_nat )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B4)),zero_zero_nat)) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( Xb = vEBT_Leaf((Uv),(Uw)) )
=> ! [N: nat] :
( ( Xaa = aa_nat_nat(suc,N) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
=> ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList,h)),l)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary),h)),one_one_nat) ) ),
one_one_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_3404_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( ( Xaa = zero_zero_nat )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( Xb = vEBT_Leaf((A4),(Uw)) )
=> ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ! [Va2: nat] :
( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList,h)),l)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary),h)),one_one_nat) ) ),
one_one_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_3405_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
Xaa = Mi,
zero_zero_nat,
$ite(
Xaa = Ma,
zero_zero_nat,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
zero_zero_nat,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
zero_zero_nat,
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma) ),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_3406_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).
% zle_diff1_eq
tff(fact_3407_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).
% zle_add1_eq_le
tff(fact_3408_vebt__member_Opelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
=> $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xaa)) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa))
=> $ite(
Xaa = Mi,
$true,
$ite(
Xaa = Ma,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
tff(fact_3409_vebt__member_Opelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
<=> (Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( (Y)
<=> $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ~ (Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
=> ( ~ (Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
=> ( ~ (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),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( (Y)
<=> $ite(
Xaa = Mi,
$true,
$ite(
Xaa = Ma,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
tff(fact_3410_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
=> $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) ) )
=> ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xaa)) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
tff(fact_3411_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_3412_double__eq__0__iff,axiom,
! [A: rat] :
( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
tff(fact_3413_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_3414_less__int__code_I1_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).
% less_int_code(1)
tff(fact_3415_times__int__code_I2_J,axiom,
! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).
% times_int_code(2)
tff(fact_3416_times__int__code_I1_J,axiom,
! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).
% times_int_code(1)
tff(fact_3417_int__distrib_I1_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(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_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(1)
tff(fact_3418_int__distrib_I2_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(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_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(2)
tff(fact_3419_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_3420_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_3421_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,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_3422_zless__add1__eq,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
| ( W = Z ) ) ) ).
% zless_add1_eq
tff(fact_3423_int__gr__induct,axiom,
! [K: int,I: int,P: fun_int_o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
=> ( ! [I3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
=> ( aa_int_o(P,I3)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
=> aa_int_o(P,I) ) ) ) ).
% int_gr_induct
tff(fact_3424_int__less__induct,axiom,
! [I: int,K: int,P: fun_int_o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
=> ( ! [I3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
=> ( aa_int_o(P,I3)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
=> aa_int_o(P,I) ) ) ) ).
% int_less_induct
tff(fact_3425_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).
% int_one_le_iff_zero_less
tff(fact_3426_pos__zmult__eq__1__iff,axiom,
! [Mb: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
<=> ( ( Mb = one_one_int )
& ( Nb = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
tff(fact_3427_odd__less__0__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(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)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).
% odd_less_0_iff
tff(fact_3428_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).
% zless_imp_add1_zle
tff(fact_3429_add1__zle__eq,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).
% add1_zle_eq
tff(fact_3430_le__imp__0__less,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).
% le_imp_0_less
tff(fact_3431_vebt__member_Opelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
=> ~ $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa))
=> ~ $ite(
Xaa = Mi,
$true,
$ite(
Xaa = Ma,
$true,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
$false,
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
$false,
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
tff(fact_3432_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
<=> (Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( (Y)
<=> $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
=> ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
=> ( ~ (Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xaa)) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
=> ( ( (Y)
<=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa)) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
tff(fact_3433_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
=> ~ $ite(
Xaa = zero_zero_nat,
(A4),
$ite(Xaa = one_one_nat,(B4),$false) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
=> ~ $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
tff(fact_3434_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
<=> (Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( ~ (Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
=> ( ~ (Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
=> ( ( (Y)
<=> ( ( Xaa = Mi )
| ( Xaa = Ma ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
=> ( ( (Y)
<=> ( ( Xaa = Mi )
| ( Xaa = Ma )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa)) ) )
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
=> ( ( (Y)
<=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
tff(fact_3435_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) )
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
=> ( ( Xaa = Mi )
| ( Xaa = Ma ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
=> ( ( Xaa = Mi )
| ( Xaa = Ma )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa))
=> $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
tff(fact_3436_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xb),Xaa) = Y )
=> ( ( ? [A4: $o,B4: $o] : Xb = vEBT_Leaf((A4),(B4))
=> ( Y != one_one_nat ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S)
=> ( Y != one_one_nat ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != $let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
& ~ ( ( Xaa = Mi )
| ( Xaa = 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,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),h),one_one_nat)),
one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_3437_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
=> ~ ( ( Xaa = Mi )
| ( Xaa = Ma ) ) ) )
=> ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
=> ~ ( ( Xaa = Mi )
| ( Xaa = Ma )
| $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
=> ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa))
=> ~ $let(
pos: nat,
pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
tff(fact_3438_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,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = $let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),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)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
& ~ ( ( Xb = Mia )
| ( Xb = Maa ) ) ),
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(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),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)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summarya),h),one_one_nat)),
one_one_nat ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_3439_cppi,axiom,
! [D5: int,P: fun_int_o,P3: fun_int_o,A3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb2: int] :
( aa_set_int_o(member_int(Xb2),A3)
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( aa_int_o(P,X4)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5)) ) )
=> ( ! [X4: int,K3: int] :
( aa_int_o(P3,X4)
<=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
=> ( ? [X_12: int] : aa_int_o(P,X_12)
<=> ( ? [X: int] :
( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
& aa_int_o(P3,X) )
| ? [X: int] :
( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
& ? [Xa2: int] :
( aa_set_int_o(member_int(Xa2),A3)
& aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X)) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_3440_cpmi,axiom,
! [D5: int,P: fun_int_o,P3: fun_int_o,B3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb2: int] :
( aa_set_int_o(member_int(Xb2),B3)
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( aa_int_o(P,X4)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5)) ) )
=> ( ! [X4: int,K3: int] :
( aa_int_o(P3,X4)
<=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
=> ( ? [X_12: int] : aa_int_o(P,X_12)
<=> ( ? [X: int] :
( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
& aa_int_o(P3,X) )
| ? [X: int] :
( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D5))
& ? [Xa2: int] :
( aa_set_int_o(member_int(Xa2),B3)
& aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X)) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_3441_bset_I6_J,axiom,
! [D5: int,B3: set_int,Ta: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),Ta) ) ) ) ).
% bset(6)
tff(fact_3442_minf_I7_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X3) ) ).
% minf(7)
tff(fact_3443_minf_I7_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X3) ) ).
% minf(7)
tff(fact_3444_minf_I7_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X3) ) ).
% minf(7)
tff(fact_3445_minf_I7_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X3) ) ).
% minf(7)
tff(fact_3446_minf_I7_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3) ) ).
% minf(7)
tff(fact_3447_minf_I5_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ta) ) ).
% minf(5)
tff(fact_3448_minf_I5_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ta) ) ).
% minf(5)
tff(fact_3449_minf_I5_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Ta) ) ).
% minf(5)
tff(fact_3450_minf_I5_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Ta) ) ).
% minf(5)
tff(fact_3451_minf_I5_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta) ) ).
% minf(5)
tff(fact_3452_minf_I4_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(4)
tff(fact_3453_minf_I4_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(4)
tff(fact_3454_minf_I4_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(4)
tff(fact_3455_minf_I4_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(4)
tff(fact_3456_minf_I4_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(4)
tff(fact_3457_minf_I3_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(3)
tff(fact_3458_minf_I3_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(3)
tff(fact_3459_minf_I3_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(3)
tff(fact_3460_minf_I3_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(3)
tff(fact_3461_minf_I3_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ( X3 != Ta ) ) ).
% minf(3)
tff(fact_3462_minf_I2_J,axiom,
! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
=> ( aa_real_o(P,X4)
<=> aa_real_o(P3,X4) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
=> ( aa_real_o(Q,X4)
<=> aa_real_o(Q6,X4) ) )
=> ? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ( ( aa_real_o(P,X3)
| aa_real_o(Q,X3) )
<=> ( aa_real_o(P3,X3)
| aa_real_o(Q6,X3) ) ) ) ) ) ).
% minf(2)
tff(fact_3463_minf_I2_J,axiom,
! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
=> ( aa_rat_o(P,X4)
<=> aa_rat_o(P3,X4) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
=> ( aa_rat_o(Q,X4)
<=> aa_rat_o(Q6,X4) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ( ( aa_rat_o(P,X3)
| aa_rat_o(Q,X3) )
<=> ( aa_rat_o(P3,X3)
| aa_rat_o(Q6,X3) ) ) ) ) ) ).
% minf(2)
tff(fact_3464_minf_I2_J,axiom,
! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
=> ( aa_num_o(P,X4)
<=> aa_num_o(P3,X4) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
=> ( aa_num_o(Q,X4)
<=> aa_num_o(Q6,X4) ) )
=> ? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> ( ( aa_num_o(P,X3)
| aa_num_o(Q,X3) )
<=> ( aa_num_o(P3,X3)
| aa_num_o(Q6,X3) ) ) ) ) ) ).
% minf(2)
tff(fact_3465_minf_I2_J,axiom,
! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
=> ( aa_nat_o(P,X4)
<=> aa_nat_o(P3,X4) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
=> ( aa_nat_o(Q,X4)
<=> aa_nat_o(Q6,X4) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ( ( aa_nat_o(P,X3)
| aa_nat_o(Q,X3) )
<=> ( aa_nat_o(P3,X3)
| aa_nat_o(Q6,X3) ) ) ) ) ) ).
% minf(2)
tff(fact_3466_minf_I2_J,axiom,
! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
=> ( aa_int_o(Q,X4)
<=> aa_int_o(Q6,X4) ) )
=> ? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ( ( aa_int_o(P,X3)
| aa_int_o(Q,X3) )
<=> ( aa_int_o(P3,X3)
| aa_int_o(Q6,X3) ) ) ) ) ) ).
% minf(2)
tff(fact_3467_minf_I1_J,axiom,
! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
=> ( aa_real_o(P,X4)
<=> aa_real_o(P3,X4) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
=> ( aa_real_o(Q,X4)
<=> aa_real_o(Q6,X4) ) )
=> ? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ( ( aa_real_o(P,X3)
& aa_real_o(Q,X3) )
<=> ( aa_real_o(P3,X3)
& aa_real_o(Q6,X3) ) ) ) ) ) ).
% minf(1)
tff(fact_3468_minf_I1_J,axiom,
! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
=> ( aa_rat_o(P,X4)
<=> aa_rat_o(P3,X4) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
=> ( aa_rat_o(Q,X4)
<=> aa_rat_o(Q6,X4) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ( ( aa_rat_o(P,X3)
& aa_rat_o(Q,X3) )
<=> ( aa_rat_o(P3,X3)
& aa_rat_o(Q6,X3) ) ) ) ) ) ).
% minf(1)
tff(fact_3469_minf_I1_J,axiom,
! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
=> ( aa_num_o(P,X4)
<=> aa_num_o(P3,X4) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
=> ( aa_num_o(Q,X4)
<=> aa_num_o(Q6,X4) ) )
=> ? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> ( ( aa_num_o(P,X3)
& aa_num_o(Q,X3) )
<=> ( aa_num_o(P3,X3)
& aa_num_o(Q6,X3) ) ) ) ) ) ).
% minf(1)
tff(fact_3470_minf_I1_J,axiom,
! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
=> ( aa_nat_o(P,X4)
<=> aa_nat_o(P3,X4) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
=> ( aa_nat_o(Q,X4)
<=> aa_nat_o(Q6,X4) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ( ( aa_nat_o(P,X3)
& aa_nat_o(Q,X3) )
<=> ( aa_nat_o(P3,X3)
& aa_nat_o(Q6,X3) ) ) ) ) ) ).
% minf(1)
tff(fact_3471_minf_I1_J,axiom,
! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
=> ( aa_int_o(Q,X4)
<=> aa_int_o(Q6,X4) ) )
=> ? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ( ( aa_int_o(P,X3)
& aa_int_o(Q,X3) )
<=> ( aa_int_o(P3,X3)
& aa_int_o(Q6,X3) ) ) ) ) ) ).
% minf(1)
tff(fact_3472_pinf_I7_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X3) ) ).
% pinf(7)
tff(fact_3473_pinf_I7_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X3) ) ).
% pinf(7)
tff(fact_3474_pinf_I7_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X3) ) ).
% pinf(7)
tff(fact_3475_pinf_I7_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X3) ) ).
% pinf(7)
tff(fact_3476_pinf_I7_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3) ) ).
% pinf(7)
tff(fact_3477_pinf_I5_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ta) ) ).
% pinf(5)
tff(fact_3478_pinf_I5_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ta) ) ).
% pinf(5)
tff(fact_3479_pinf_I5_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Ta) ) ).
% pinf(5)
tff(fact_3480_pinf_I5_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Ta) ) ).
% pinf(5)
tff(fact_3481_pinf_I5_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta) ) ).
% pinf(5)
tff(fact_3482_pinf_I4_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(4)
tff(fact_3483_pinf_I4_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(4)
tff(fact_3484_pinf_I4_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(4)
tff(fact_3485_pinf_I4_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(4)
tff(fact_3486_pinf_I4_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(4)
tff(fact_3487_pinf_I3_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(3)
tff(fact_3488_pinf_I3_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(3)
tff(fact_3489_pinf_I3_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(3)
tff(fact_3490_pinf_I3_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(3)
tff(fact_3491_pinf_I3_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ( X3 != Ta ) ) ).
% pinf(3)
tff(fact_3492_pinf_I2_J,axiom,
! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
=> ( aa_real_o(P,X4)
<=> aa_real_o(P3,X4) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
=> ( aa_real_o(Q,X4)
<=> aa_real_o(Q6,X4) ) )
=> ? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ( ( aa_real_o(P,X3)
| aa_real_o(Q,X3) )
<=> ( aa_real_o(P3,X3)
| aa_real_o(Q6,X3) ) ) ) ) ) ).
% pinf(2)
tff(fact_3493_pinf_I2_J,axiom,
! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
=> ( aa_rat_o(P,X4)
<=> aa_rat_o(P3,X4) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
=> ( aa_rat_o(Q,X4)
<=> aa_rat_o(Q6,X4) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ( ( aa_rat_o(P,X3)
| aa_rat_o(Q,X3) )
<=> ( aa_rat_o(P3,X3)
| aa_rat_o(Q6,X3) ) ) ) ) ) ).
% pinf(2)
tff(fact_3494_pinf_I2_J,axiom,
! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
=> ( aa_num_o(P,X4)
<=> aa_num_o(P3,X4) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
=> ( aa_num_o(Q,X4)
<=> aa_num_o(Q6,X4) ) )
=> ? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> ( ( aa_num_o(P,X3)
| aa_num_o(Q,X3) )
<=> ( aa_num_o(P3,X3)
| aa_num_o(Q6,X3) ) ) ) ) ) ).
% pinf(2)
tff(fact_3495_pinf_I2_J,axiom,
! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
=> ( aa_nat_o(P,X4)
<=> aa_nat_o(P3,X4) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
=> ( aa_nat_o(Q,X4)
<=> aa_nat_o(Q6,X4) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ( ( aa_nat_o(P,X3)
| aa_nat_o(Q,X3) )
<=> ( aa_nat_o(P3,X3)
| aa_nat_o(Q6,X3) ) ) ) ) ) ).
% pinf(2)
tff(fact_3496_pinf_I2_J,axiom,
! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
=> ( aa_int_o(Q,X4)
<=> aa_int_o(Q6,X4) ) )
=> ? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ( ( aa_int_o(P,X3)
| aa_int_o(Q,X3) )
<=> ( aa_int_o(P3,X3)
| aa_int_o(Q6,X3) ) ) ) ) ) ).
% pinf(2)
tff(fact_3497_pinf_I1_J,axiom,
! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
=> ( aa_real_o(P,X4)
<=> aa_real_o(P3,X4) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
=> ( aa_real_o(Q,X4)
<=> aa_real_o(Q6,X4) ) )
=> ? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ( ( aa_real_o(P,X3)
& aa_real_o(Q,X3) )
<=> ( aa_real_o(P3,X3)
& aa_real_o(Q6,X3) ) ) ) ) ) ).
% pinf(1)
tff(fact_3498_pinf_I1_J,axiom,
! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
=> ( aa_rat_o(P,X4)
<=> aa_rat_o(P3,X4) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
=> ( aa_rat_o(Q,X4)
<=> aa_rat_o(Q6,X4) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ( ( aa_rat_o(P,X3)
& aa_rat_o(Q,X3) )
<=> ( aa_rat_o(P3,X3)
& aa_rat_o(Q6,X3) ) ) ) ) ) ).
% pinf(1)
tff(fact_3499_pinf_I1_J,axiom,
! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
=> ( aa_num_o(P,X4)
<=> aa_num_o(P3,X4) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
=> ( aa_num_o(Q,X4)
<=> aa_num_o(Q6,X4) ) )
=> ? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> ( ( aa_num_o(P,X3)
& aa_num_o(Q,X3) )
<=> ( aa_num_o(P3,X3)
& aa_num_o(Q6,X3) ) ) ) ) ) ).
% pinf(1)
tff(fact_3500_pinf_I1_J,axiom,
! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
=> ( aa_nat_o(P,X4)
<=> aa_nat_o(P3,X4) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
=> ( aa_nat_o(Q,X4)
<=> aa_nat_o(Q6,X4) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ( ( aa_nat_o(P,X3)
& aa_nat_o(Q,X3) )
<=> ( aa_nat_o(P3,X3)
& aa_nat_o(Q6,X3) ) ) ) ) ) ).
% pinf(1)
tff(fact_3501_pinf_I1_J,axiom,
! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
=> ( aa_int_o(Q,X4)
<=> aa_int_o(Q6,X4) ) )
=> ? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ( ( aa_int_o(P,X3)
& aa_int_o(Q,X3) )
<=> ( aa_int_o(P3,X3)
& aa_int_o(Q6,X3) ) ) ) ) ) ).
% pinf(1)
tff(fact_3502_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: $o,B: $o,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A),(B))),Xb) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_3503_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,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero_nat,Ts,S2)),Xb) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_3504_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,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2)),Xb) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_3505_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,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: 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)),TreeLista,Summarya)),Xb) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_3506_insersimp_H,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_13)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),Y)),one_one_nat) ) ) ).
% insersimp'
tff(fact_3507_pinf_I6_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Ta) ) ).
% pinf(6)
tff(fact_3508_pinf_I6_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Ta) ) ).
% pinf(6)
tff(fact_3509_pinf_I6_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ta) ) ).
% pinf(6)
tff(fact_3510_pinf_I6_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta) ) ).
% pinf(6)
tff(fact_3511_pinf_I6_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ta) ) ).
% pinf(6)
tff(fact_3512_pinf_I8_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X3) ) ).
% pinf(8)
tff(fact_3513_pinf_I8_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X3) ) ).
% pinf(8)
tff(fact_3514_pinf_I8_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X3) ) ).
% pinf(8)
tff(fact_3515_pinf_I8_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3) ) ).
% pinf(8)
tff(fact_3516_pinf_I8_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X3) ) ).
% pinf(8)
tff(fact_3517_minf_I6_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Ta) ) ).
% minf(6)
tff(fact_3518_minf_I6_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Ta) ) ).
% minf(6)
tff(fact_3519_minf_I6_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ta) ) ).
% minf(6)
tff(fact_3520_minf_I6_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta) ) ).
% minf(6)
tff(fact_3521_minf_I6_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ta) ) ).
% minf(6)
tff(fact_3522_minf_I8_J,axiom,
! [Ta: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X3) ) ).
% minf(8)
tff(fact_3523_minf_I8_J,axiom,
! [Ta: num] :
? [Z2: num] :
! [X3: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X3) ) ).
% minf(8)
tff(fact_3524_minf_I8_J,axiom,
! [Ta: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X3) ) ).
% minf(8)
tff(fact_3525_minf_I8_J,axiom,
! [Ta: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3) ) ).
% minf(8)
tff(fact_3526_minf_I8_J,axiom,
! [Ta: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X3) ) ).
% minf(8)
tff(fact_3527_insertsimp_H,axiom,
! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),L)),one_one_nat) ) ) ).
% insertsimp'
tff(fact_3528_inf__period_I2_J,axiom,
! [P: fun_real_o,D5: real,Q: fun_real_o] :
( ! [X4: real,K3: real] :
( aa_real_o(P,X4)
<=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
=> ( ! [X4: real,K3: real] :
( aa_real_o(Q,X4)
<=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
=> ! [X3: real,K4: real] :
( ( aa_real_o(P,X3)
| aa_real_o(Q,X3) )
<=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5)))
| aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))) ) ) ) ) ).
% inf_period(2)
tff(fact_3529_inf__period_I2_J,axiom,
! [P: fun_rat_o,D5: rat,Q: fun_rat_o] :
( ! [X4: rat,K3: rat] :
( aa_rat_o(P,X4)
<=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K3),D5))) )
=> ( ! [X4: rat,K3: rat] :
( aa_rat_o(Q,X4)
<=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K3),D5))) )
=> ! [X3: rat,K4: rat] :
( ( aa_rat_o(P,X3)
| aa_rat_o(Q,X3) )
<=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5)))
| aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5))) ) ) ) ) ).
% inf_period(2)
tff(fact_3530_inf__period_I2_J,axiom,
! [P: fun_int_o,D5: int,Q: fun_int_o] :
( ! [X4: int,K3: int] :
( aa_int_o(P,X4)
<=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
=> ( ! [X4: int,K3: int] :
( aa_int_o(Q,X4)
<=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
=> ! [X3: int,K4: int] :
( ( aa_int_o(P,X3)
| aa_int_o(Q,X3) )
<=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5)))
| aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))) ) ) ) ) ).
% inf_period(2)
tff(fact_3531_inf__period_I1_J,axiom,
! [P: fun_real_o,D5: real,Q: fun_real_o] :
( ! [X4: real,K3: real] :
( aa_real_o(P,X4)
<=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
=> ( ! [X4: real,K3: real] :
( aa_real_o(Q,X4)
<=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D5))) )
=> ! [X3: real,K4: real] :
( ( aa_real_o(P,X3)
& aa_real_o(Q,X3) )
<=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5)))
& aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))) ) ) ) ) ).
% inf_period(1)
tff(fact_3532_inf__period_I1_J,axiom,
! [P: fun_rat_o,D5: rat,Q: fun_rat_o] :
( ! [X4: rat,K3: rat] :
( aa_rat_o(P,X4)
<=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K3),D5))) )
=> ( ! [X4: rat,K3: rat] :
( aa_rat_o(Q,X4)
<=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K3),D5))) )
=> ! [X3: rat,K4: rat] :
( ( aa_rat_o(P,X3)
& aa_rat_o(Q,X3) )
<=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5)))
& aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5))) ) ) ) ) ).
% inf_period(1)
tff(fact_3533_inf__period_I1_J,axiom,
! [P: fun_int_o,D5: int,Q: fun_int_o] :
( ! [X4: int,K3: int] :
( aa_int_o(P,X4)
<=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
=> ( ! [X4: int,K3: int] :
( aa_int_o(Q,X4)
<=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D5))) )
=> ! [X3: int,K4: int] :
( ( aa_int_o(P,X3)
& aa_int_o(Q,X3) )
<=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5)))
& aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))) ) ) ) ) ).
% inf_period(1)
tff(fact_3534_insert_H__bound__height,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).
% insert'_bound_height
tff(fact_3535_plusinfinity,axiom,
! [D: int,P3: fun_int_o,P: fun_int_o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> ( ! [X4: int,K3: int] :
( aa_int_o(P3,X4)
<=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P3,X4) ) )
=> ( ? [X_1: int] : aa_int_o(P3,X_1)
=> ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).
% plusinfinity
tff(fact_3536_minusinfinity,axiom,
! [D: int,P1: fun_int_o,P: fun_int_o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> ( ! [X4: int,K3: int] :
( aa_int_o(P1,X4)
<=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
=> ( ? [Z4: int] :
! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
=> ( aa_int_o(P,X4)
<=> aa_int_o(P1,X4) ) )
=> ( ? [X_1: int] : aa_int_o(P1,X_1)
=> ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).
% minusinfinity
tff(fact_3537_incr__mult__lemma,axiom,
! [D: int,P: fun_int_o,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> ( ! [X4: int] :
( aa_int_o(P,X4)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D)) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ! [X3: int] :
( aa_int_o(P,X3)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).
% incr_mult_lemma
tff(fact_3538_decr__mult__lemma,axiom,
! [D: int,P: fun_int_o,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> ( ! [X4: int] :
( aa_int_o(P,X4)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D)) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ! [X3: int] :
( aa_int_o(P,X3)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).
% decr_mult_lemma
tff(fact_3539_periodic__finite__ex,axiom,
! [D: int,P: fun_int_o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> ( ! [X4: int,K3: int] :
( aa_int_o(P,X4)
<=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
=> ( ? [X_12: int] : aa_int_o(P,X_12)
<=> ? [X: int] :
( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D))
& aa_int_o(P,X) ) ) ) ) ).
% periodic_finite_ex
tff(fact_3540_bset_I3_J,axiom,
! [D5: int,Ta: int,B3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ( X3 = Ta )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5) = Ta ) ) ) ) ) ).
% bset(3)
tff(fact_3541_bset_I4_J,axiom,
! [D5: int,Ta: int,B3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(Ta),B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ( X3 != Ta )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5) != Ta ) ) ) ) ) ).
% bset(4)
tff(fact_3542_bset_I5_J,axiom,
! [D5: int,B3: set_int,Ta: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),Ta) ) ) ) ).
% bset(5)
tff(fact_3543_bset_I7_J,axiom,
! [D5: int,Ta: int,B3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(Ta),B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) ) ) ) ).
% bset(7)
tff(fact_3544_aset_I3_J,axiom,
! [D5: int,Ta: int,A3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ( X3 = Ta )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5) = Ta ) ) ) ) ) ).
% aset(3)
tff(fact_3545_aset_I4_J,axiom,
! [D5: int,Ta: int,A3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(Ta),A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ( X3 != Ta )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5) != Ta ) ) ) ) ) ).
% aset(4)
tff(fact_3546_aset_I5_J,axiom,
! [D5: int,Ta: int,A3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(Ta),A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),Ta) ) ) ) ) ).
% aset(5)
tff(fact_3547_aset_I7_J,axiom,
! [D5: int,A3: set_int,Ta: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) ) ) ).
% aset(7)
tff(fact_3548_aset_I8_J,axiom,
! [D5: int,A3: set_int,Ta: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) ) ) ).
% aset(8)
tff(fact_3549_aset_I6_J,axiom,
! [D5: int,Ta: int,A3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),Ta) ) ) ) ) ).
% aset(6)
tff(fact_3550_bset_I8_J,axiom,
! [D5: int,Ta: int,B3: set_int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
=> ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5))
=> ! [Xb3: int] :
( aa_set_int_o(member_int(Xb3),B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) ) ) ) ).
% bset(8)
tff(fact_3551_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),Xaa)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xaa)) ) )
=> ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
=> ( ( 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)),TreeList,Summary),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = $let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
& ~ ( ( Xaa = Mi )
| ( Xaa = 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,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),h),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(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_3552_psubsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
=> ( ( A3 != B3 )
=> aa_set_nat_o(ord_less_set_nat(A3),B3) ) ) ).
% psubsetI
tff(fact_3553_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_3554_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_3555_Bolzano,axiom,
! [A: real,B: real,P: fun_real_fun_real_o] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( ! [A4: real,B4: real,C3: real] :
( aa_real_o(aa_real_fun_real_o(P,A4),B4)
=> ( aa_real_o(aa_real_fun_real_o(P,B4),C3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),B4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),C3)
=> aa_real_o(aa_real_fun_real_o(P,A4),C3) ) ) ) )
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
=> ? [D6: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D6)
& ! [A4: real,B4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A4)),D6) )
=> aa_real_o(aa_real_fun_real_o(P,A4),B4) ) ) ) )
=> aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).
% Bolzano
tff(fact_3556_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y )
=> ( ! [A4: $o] :
( ? [B4: $o] : Xb = vEBT_Leaf((A4),(B4))
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((A4),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( Y != one_one_nat ) )
=> ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
=> ( Y != one_one_nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_3557_mult__le__cancel__iff1,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).
% mult_le_cancel_iff1
tff(fact_3558_mult__le__cancel__iff1,axiom,
! [Z: int,Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).
% mult_le_cancel_iff1
tff(fact_3559_mult__le__cancel__iff1,axiom,
! [Z: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).
% mult_le_cancel_iff1
tff(fact_3560_unset__bit__negative__int__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% unset_bit_negative_int_iff
tff(fact_3561_psubsetD,axiom,
! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ord_le7866589430770878221at_nat(A3,B3)
=> ( member8440522571783428010at_nat(C,A3)
=> member8440522571783428010at_nat(C,B3) ) ) ).
% psubsetD
tff(fact_3562_psubsetD,axiom,
! [A3: set_complex,B3: set_complex,C: complex] :
( aa_set_complex_o(ord_less_set_complex(A3),B3)
=> ( aa_set_complex_o(member_complex(C),A3)
=> aa_set_complex_o(member_complex(C),B3) ) ) ).
% psubsetD
tff(fact_3563_psubsetD,axiom,
! [A3: set_real,B3: set_real,C: real] :
( aa_set_real_o(ord_less_set_real(A3),B3)
=> ( aa_set_real_o(member_real(C),A3)
=> aa_set_real_o(member_real(C),B3) ) ) ).
% psubsetD
tff(fact_3564_psubsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
=> ( aa_set_nat_o(member_nat(C),A3)
=> aa_set_nat_o(member_nat(C),B3) ) ) ).
% psubsetD
tff(fact_3565_psubsetD,axiom,
! [A3: set_int,B3: set_int,C: int] :
( aa_set_int_o(ord_less_set_int(A3),B3)
=> ( aa_set_int_o(member_int(C),A3)
=> aa_set_int_o(member_int(C),B3) ) ) ).
% psubsetD
tff(fact_3566_psubset__imp__ex__mem,axiom,
! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ord_le7866589430770878221at_nat(A3,B3)
=> ? [B4: product_prod_nat_nat] : member8440522571783428010at_nat(B4,minus_1356011639430497352at_nat(B3,A3)) ) ).
% psubset_imp_ex_mem
tff(fact_3567_psubset__imp__ex__mem,axiom,
! [A3: set_complex,B3: set_complex] :
( aa_set_complex_o(ord_less_set_complex(A3),B3)
=> ? [B4: complex] : aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3)) ) ).
% psubset_imp_ex_mem
tff(fact_3568_psubset__imp__ex__mem,axiom,
! [A3: set_real,B3: set_real] :
( aa_set_real_o(ord_less_set_real(A3),B3)
=> ? [B4: real] : aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3)) ) ).
% psubset_imp_ex_mem
tff(fact_3569_psubset__imp__ex__mem,axiom,
! [A3: set_int,B3: set_int] :
( aa_set_int_o(ord_less_set_int(A3),B3)
=> ? [B4: int] : aa_set_int_o(member_int(B4),minus_minus_set_int(B3,A3)) ) ).
% psubset_imp_ex_mem
tff(fact_3570_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
=> ? [B4: nat] : aa_set_nat_o(member_nat(B4),minus_minus_set_nat(B3,A3)) ) ).
% psubset_imp_ex_mem
tff(fact_3571_less__set__def,axiom,
! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ord_le7866589430770878221at_nat(A3,B3)
<=> aa_fun8408151826566693824at_o_o(ord_le9073604597836486254_nat_o(aTP_Lamm_a(A3)),aTP_Lamm_a(B3)) ) ).
% less_set_def
tff(fact_3572_less__set__def,axiom,
! [A3: set_complex,B3: set_complex] :
( aa_set_complex_o(ord_less_set_complex(A3),B3)
<=> ord_le4409593196786536943plex_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B3)) ) ).
% less_set_def
tff(fact_3573_less__set__def,axiom,
! [A3: set_real,B3: set_real] :
( aa_set_real_o(ord_less_set_real(A3),B3)
<=> ord_less_fun_real_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3)) ) ).
% less_set_def
tff(fact_3574_less__set__def,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
<=> ord_less_fun_nat_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3)) ) ).
% less_set_def
tff(fact_3575_less__set__def,axiom,
! [A3: set_int,B3: set_int] :
( aa_set_int_o(ord_less_set_int(A3),B3)
<=> ord_less_fun_int_o(aTP_Lamm_af(A3),aTP_Lamm_af(B3)) ) ).
% less_set_def
tff(fact_3576_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_3577_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A: $o,B: $o] :
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),
$ite((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_3578_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mia: nat,Maa: 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(Mia),Maa)),Ux,Uy,Uz)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_3579_not__psubset__empty,axiom,
! [A3: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A3),bot_bot_set_nat) ).
% not_psubset_empty
tff(fact_3580_not__psubset__empty,axiom,
! [A3: set_int] : ~ aa_set_int_o(ord_less_set_int(A3),bot_bot_set_int) ).
% not_psubset_empty
tff(fact_3581_not__psubset__empty,axiom,
! [A3: set_real] : ~ aa_set_real_o(ord_less_set_real(A3),bot_bot_set_real) ).
% not_psubset_empty
tff(fact_3582_psubsetE,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
=> ~ ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
=> aa_set_nat_o(ord_less_eq_set_nat(B3),A3) ) ) ).
% psubsetE
tff(fact_3583_psubset__eq,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
& ( A3 != B3 ) ) ) ).
% psubset_eq
tff(fact_3584_psubset__imp__subset,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
=> aa_set_nat_o(ord_less_eq_set_nat(A3),B3) ) ).
% psubset_imp_subset
tff(fact_3585_psubset__subset__trans,axiom,
! [A3: set_nat,B3: set_nat,C4: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(B3),C4)
=> aa_set_nat_o(ord_less_set_nat(A3),C4) ) ) ).
% psubset_subset_trans
tff(fact_3586_subset__not__subset__eq,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),B3)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
& ~ aa_set_nat_o(ord_less_eq_set_nat(B3),A3) ) ) ).
% subset_not_subset_eq
tff(fact_3587_subset__psubset__trans,axiom,
! [A3: set_nat,B3: set_nat,C4: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
=> ( aa_set_nat_o(ord_less_set_nat(B3),C4)
=> aa_set_nat_o(ord_less_set_nat(A3),C4) ) ) ).
% subset_psubset_trans
tff(fact_3588_subset__iff__psubset__eq,axiom,
! [A3: set_nat,B3: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
<=> ( aa_set_nat_o(ord_less_set_nat(A3),B3)
| ( A3 = B3 ) ) ) ).
% subset_iff_psubset_eq
tff(fact_3589_mult__less__iff1,axiom,
! [Z: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).
% mult_less_iff1
tff(fact_3590_mult__less__iff1,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).
% mult_less_iff1
tff(fact_3591_mult__less__iff1,axiom,
! [Z: int,Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).
% mult_less_iff1
tff(fact_3592_mult__le__cancel__iff2,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xb)),aa_rat_rat(times_times_rat(Z),Y))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).
% mult_le_cancel_iff2
tff(fact_3593_mult__le__cancel__iff2,axiom,
! [Z: int,Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).
% mult_le_cancel_iff2
tff(fact_3594_mult__le__cancel__iff2,axiom,
! [Z: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).
% mult_le_cancel_iff2
tff(fact_3595_divides__aux__eq,axiom,
! [Q3: nat,R2: nat] :
( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2))
<=> ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
tff(fact_3596_divides__aux__eq,axiom,
! [Q3: int,R2: int] :
( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
<=> ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
tff(fact_3597_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q3: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> ( aa_Pro4592068669816069595_int_o(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,Q3),R2))
=> aa_Pro4592068669816069595_int_o(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,Q3),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))),R2)),one_one_int))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_3598_low__def,axiom,
! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% low_def
tff(fact_3599_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((B4),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( Y != one_one_nat ) )
=> ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
=> ( Y != one_one_nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_3600_obtain__set__succ,axiom,
! [Xb: nat,Z: nat,A3: set_nat,B3: set_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
=> ( aa_nat_o(vEBT_VEBT_max_in_set(A3),Z)
=> ( aa_set_nat_o(finite_finite_nat,B3)
=> ( ( A3 = B3 )
=> ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A3),Xb),X_13) ) ) ) ) ).
% obtain_set_succ
tff(fact_3601_obtain__set__pred,axiom,
! [Z: nat,Xb: nat,A3: set_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
=> ( aa_nat_o(vEBT_VEBT_min_in_set(A3),Z)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A3),Xb),X_13) ) ) ) ).
% obtain_set_pred
tff(fact_3602_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_s_u_c_c(Xb),Xaa) = Y )
=> ( ( ? [Uu: $o,B4: $o] : Xb = vEBT_Leaf((Uu),(B4))
=> ( ( Xaa = zero_zero_nat )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
=> ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
=> ( Y != one_one_nat ) ) )
=> ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
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(bit0,one))))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,h)))),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList,h)),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summary,h),
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,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary),h))),one_one_nat)),
$ite(sc = none_nat,one_one_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(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(sc)))))) ) )) )),
one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_3603_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_p_r_e_d(Xb),Xaa) = Y )
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> ( ( Xaa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ? [A4: $o,Uw: $o] : Xb = vEBT_Leaf((A4),(Uw))
=> ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((B4),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
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(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList,h)),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summary,h),
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,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary),h))),one_one_nat)),
$ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_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(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(pr)))))) ) )) ),
one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_3604_set__vebt__finite,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).
% set_vebt_finite
tff(fact_3605_pred__none__empty,axiom,
! [Xs: set_nat,A: nat] :
( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_13)
=> ( aa_set_nat_o(finite_finite_nat,Xs)
=> ~ ? [X3: nat] :
( aa_set_nat_o(member_nat(X3),Xs)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),A) ) ) ) ).
% pred_none_empty
tff(fact_3606_succ__none__empty,axiom,
! [Xs: set_nat,A: nat] :
( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_13)
=> ( aa_set_nat_o(finite_finite_nat,Xs)
=> ~ ? [X3: nat] :
( aa_set_nat_o(member_nat(X3),Xs)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X3) ) ) ) ).
% succ_none_empty
tff(fact_3607_semiring__norm_I90_J,axiom,
! [Mb: num,Nb: num] :
( ( aa_num_num(bit1,Mb) = aa_num_num(bit1,Nb) )
<=> ( Mb = Nb ) ) ).
% semiring_norm(90)
tff(fact_3608_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
<=> ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
tff(fact_3609_semiring__norm_I89_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).
% semiring_norm(89)
tff(fact_3610_semiring__norm_I88_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).
% semiring_norm(88)
tff(fact_3611_semiring__norm_I86_J,axiom,
! [Mb: num] : aa_num_num(bit1,Mb) != one ).
% semiring_norm(86)
tff(fact_3612_semiring__norm_I84_J,axiom,
! [Nb: num] : one != aa_num_num(bit1,Nb) ).
% semiring_norm(84)
tff(fact_3613_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_3614_mod__add__self2,axiom,
! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).
% mod_add_self2
tff(fact_3615_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_3616_mod__add__self1,axiom,
! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).
% mod_add_self1
tff(fact_3617_List_Ofinite__set,axiom,
! [Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs)) ).
% List.finite_set
tff(fact_3618_List_Ofinite__set,axiom,
! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).
% List.finite_set
tff(fact_3619_List_Ofinite__set,axiom,
! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).
% List.finite_set
tff(fact_3620_List_Ofinite__set,axiom,
! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).
% List.finite_set
tff(fact_3621_mod__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).
% mod_less
tff(fact_3622_semiring__norm_I73_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% semiring_norm(73)
tff(fact_3623_semiring__norm_I80_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% semiring_norm(80)
tff(fact_3624_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_3625_mod__mult__self1__is__0,axiom,
! [B: int,A: int] : aa_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_3626_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_3627_mod__mult__self2__is__0,axiom,
! [A: int,B: int] : aa_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_3628_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_3629_bits__mod__by__1,axiom,
! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).
% bits_mod_by_1
tff(fact_3630_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_3631_mod__by__1,axiom,
! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).
% mod_by_1
tff(fact_3632_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_3633_mod__mult__self1,axiom,
! [A: int,C: int,B: int] : aa_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(modulo_modulo_int(A),B) ).
% mod_mult_self1
tff(fact_3634_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_3635_mod__mult__self2,axiom,
! [A: int,B: int,C: int] : aa_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(modulo_modulo_int(A),B) ).
% mod_mult_self2
tff(fact_3636_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_3637_mod__mult__self3,axiom,
! [C: int,B: int,A: int] : aa_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(modulo_modulo_int(A),B) ).
% mod_mult_self3
tff(fact_3638_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_3639_mod__mult__self4,axiom,
! [B: int,C: int,A: int] : aa_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(modulo_modulo_int(A),B) ).
% mod_mult_self4
tff(fact_3640_infinite__Icc__iff,axiom,
! [A: rat,B: rat] :
( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% infinite_Icc_iff
tff(fact_3641_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% infinite_Icc_iff
tff(fact_3642_mod__by__Suc__0,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
% mod_by_Suc_0
tff(fact_3643_semiring__norm_I9_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% semiring_norm(9)
tff(fact_3644_semiring__norm_I7_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% semiring_norm(7)
tff(fact_3645_semiring__norm_I14_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),aa_num_num(bit1,Nb))) ).
% semiring_norm(14)
tff(fact_3646_semiring__norm_I15_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),Nb)) ).
% semiring_norm(15)
tff(fact_3647_semiring__norm_I72_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% semiring_norm(72)
tff(fact_3648_semiring__norm_I81_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% semiring_norm(81)
tff(fact_3649_semiring__norm_I70_J,axiom,
! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).
% semiring_norm(70)
tff(fact_3650_semiring__norm_I77_J,axiom,
! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).
% semiring_norm(77)
tff(fact_3651_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] : aa_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(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).
% zdiv_numeral_Bit1
tff(fact_3652_Suc__mod__mult__self4,axiom,
! [Nb: nat,K: nat,Mb: 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,Nb),K)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).
% Suc_mod_mult_self4
tff(fact_3653_Suc__mod__mult__self3,axiom,
! [K: nat,Nb: nat,Mb: 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),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).
% Suc_mod_mult_self3
tff(fact_3654_Suc__mod__mult__self2,axiom,
! [Mb: nat,Nb: 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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).
% Suc_mod_mult_self2
tff(fact_3655_Suc__mod__mult__self1,axiom,
! [Mb: nat,K: nat,Nb: 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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).
% Suc_mod_mult_self1
tff(fact_3656_semiring__norm_I10_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = 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,Mb),Nb)),one)) ).
% semiring_norm(10)
tff(fact_3657_semiring__norm_I8_J,axiom,
! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one)) ).
% semiring_norm(8)
tff(fact_3658_semiring__norm_I5_J,axiom,
! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).
% semiring_norm(5)
tff(fact_3659_semiring__norm_I4_J,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).
% semiring_norm(4)
tff(fact_3660_semiring__norm_I3_J,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).
% semiring_norm(3)
tff(fact_3661_semiring__norm_I16_J,axiom,
! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = 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,Mb),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)))) ).
% semiring_norm(16)
tff(fact_3662_semiring__norm_I74_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% semiring_norm(74)
tff(fact_3663_semiring__norm_I79_J,axiom,
! [Mb: num,Nb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).
% semiring_norm(79)
tff(fact_3664_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_3665_bits__one__mod__two__eq__one,axiom,
aa_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_3666_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_3667_one__mod__two__eq__one,axiom,
aa_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_3668_mod2__Suc__Suc,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% mod2_Suc_Suc
tff(fact_3669_Suc__times__numeral__mod__eq,axiom,
! [K: num,Nb: 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)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
tff(fact_3670_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_3671_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
<=> ( aa_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_3672_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_3673_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
<=> ( aa_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_3674_not__mod2__eq__Suc__0__eq__0,axiom,
! [Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
tff(fact_3675_div__Suc__eq__div__add3,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).
% div_Suc_eq_div_add3
tff(fact_3676_Suc__div__eq__add3__div__numeral,axiom,
! [Mb: 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,Mb)))),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))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).
% Suc_div_eq_add3_div_numeral
tff(fact_3677_add__self__mod__2,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% add_self_mod_2
tff(fact_3678_mod__Suc__eq__mod__add3,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).
% mod_Suc_eq_mod_add3
tff(fact_3679_Suc__mod__eq__add3__mod__numeral,axiom,
! [Mb: 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,Mb)))),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))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).
% Suc_mod_eq_add3_mod_numeral
tff(fact_3680_mod2__gr__0,axiom,
! [Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).
% mod2_gr_0
tff(fact_3681_cong__exp__iff__simps_I13_J,axiom,
! [Mb: num,Q3: num,Nb: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).
% cong_exp_iff_simps(13)
tff(fact_3682_cong__exp__iff__simps_I13_J,axiom,
! [Mb: num,Q3: num,Nb: num] :
( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
<=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).
% cong_exp_iff_simps(13)
tff(fact_3683_cong__exp__iff__simps_I12_J,axiom,
! [Mb: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(12)
tff(fact_3684_cong__exp__iff__simps_I12_J,axiom,
! [Mb: num,Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(12)
tff(fact_3685_cong__exp__iff__simps_I10_J,axiom,
! [Mb: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(10)
tff(fact_3686_cong__exp__iff__simps_I10_J,axiom,
! [Mb: num,Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(10)
tff(fact_3687_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_3688_mod__mult__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_eq
tff(fact_3689_mod__mult__cong,axiom,
! [A: nat,C: nat,A2: nat,B: nat,B2: 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,A2),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,B2),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,A2),B2)),C) ) ) ) ).
% mod_mult_cong
tff(fact_3690_mod__mult__cong,axiom,
! [A: int,C: int,A2: int,B: int,B2: int] :
( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A2),C) )
=> ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B2),C) )
=> ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ) ) ) ).
% mod_mult_cong
tff(fact_3691_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_3692_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] : aa_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(modulo_modulo_int(A),B)),C) ).
% mod_mult_mult2
tff(fact_3693_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_3694_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(modulo_modulo_int(A),B)) = aa_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_3695_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_3696_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_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_3697_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_3698_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_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_3699_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_3700_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_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_3701_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_3702_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_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_3703_mod__add__cong,axiom,
! [A: nat,C: nat,A2: nat,B: nat,B2: 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,A2),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,B2),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,A2),B2)),C) ) ) ) ).
% mod_add_cong
tff(fact_3704_mod__add__cong,axiom,
! [A: int,C: int,A2: int,B: int,B2: int] :
( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A2),C) )
=> ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B2),C) )
=> ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ) ) ) ).
% mod_add_cong
tff(fact_3705_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_3706_mod__add__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_eq
tff(fact_3707_power__mod,axiom,
! [A: nat,B: nat,Nb: 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)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).
% power_mod
tff(fact_3708_power__mod,axiom,
! [A: int,B: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),Nb)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),Nb)),B) ).
% power_mod
tff(fact_3709_mod__Suc__eq,axiom,
! [Mb: nat,Nb: 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,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).
% mod_Suc_eq
tff(fact_3710_mod__Suc__Suc__eq,axiom,
! [Mb: nat,Nb: 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,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).
% mod_Suc_Suc_eq
tff(fact_3711_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).
% verit_eq_simplify(14)
tff(fact_3712_mod__less__eq__dividend,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).
% mod_less_eq_dividend
tff(fact_3713_verit__eq__simplify_I12_J,axiom,
! [X32: num] : one != aa_num_num(bit1,X32) ).
% verit_eq_simplify(12)
tff(fact_3714_finite__nat__set__iff__bounded,axiom,
! [N4: set_nat] :
( aa_set_nat_o(finite_finite_nat,N4)
<=> ? [M4: nat] :
! [X: nat] :
( aa_set_nat_o(member_nat(X),N4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),M4) ) ) ).
% finite_nat_set_iff_bounded
tff(fact_3715_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,Nb: nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),N4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Nb) )
=> aa_set_nat_o(finite_finite_nat,N4) ) ).
% bounded_nat_set_is_finite
tff(fact_3716_finite__nat__set__iff__bounded__le,axiom,
! [N4: set_nat] :
( aa_set_nat_o(finite_finite_nat,N4)
<=> ? [M4: nat] :
! [X: nat] :
( aa_set_nat_o(member_nat(X),N4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M4) ) ) ).
% finite_nat_set_iff_bounded_le
tff(fact_3717_finite__list,axiom,
! [A3: set_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A3 ) ).
% finite_list
tff(fact_3718_finite__list,axiom,
! [A3: set_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ? [Xs2: list_nat] : set_nat2(Xs2) = A3 ) ).
% finite_list
tff(fact_3719_finite__list,axiom,
! [A3: set_int] :
( aa_set_int_o(finite_finite_int,A3)
=> ? [Xs2: list_int] : set_int2(Xs2) = A3 ) ).
% finite_list
tff(fact_3720_finite__list,axiom,
! [A3: set_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ? [Xs2: list_complex] : set_complex2(Xs2) = A3 ) ).
% finite_list
tff(fact_3721_cong__exp__iff__simps_I3_J,axiom,
! [Nb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).
% cong_exp_iff_simps(3)
tff(fact_3722_cong__exp__iff__simps_I3_J,axiom,
! [Nb: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).
% cong_exp_iff_simps(3)
tff(fact_3723_finite__M__bounded__by__nat,axiom,
! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aw(P),I))) ).
% finite_M_bounded_by_nat
tff(fact_3724_finite__less__ub,axiom,
! [F: fun_nat_nat,U: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ax(F),U))) ) ).
% finite_less_ub
tff(fact_3725_finite__lists__length__eq,axiom,
! [A3: set_complex,Nb: nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_ay(A3),Nb))) ) ).
% finite_lists_length_eq
tff(fact_3726_finite__lists__length__eq,axiom,
! [A3: set_VEBT_VEBT,Nb: nat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_az(A3),Nb))) ) ).
% finite_lists_length_eq
tff(fact_3727_finite__lists__length__eq,axiom,
! [A3: set_o,Nb: nat] :
( finite_finite_o(A3)
=> finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_ba(A3),Nb))) ) ).
% finite_lists_length_eq
tff(fact_3728_finite__lists__length__eq,axiom,
! [A3: set_int,Nb: nat] :
( aa_set_int_o(finite_finite_int,A3)
=> finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bb(A3),Nb))) ) ).
% finite_lists_length_eq
tff(fact_3729_finite__lists__length__eq,axiom,
! [A3: set_nat,Nb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bc(A3),Nb))) ) ).
% finite_lists_length_eq
tff(fact_3730_cong__exp__iff__simps_I11_J,axiom,
! [Mb: num,Q3: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = 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,Q3))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
tff(fact_3731_cong__exp__iff__simps_I11_J,axiom,
! [Mb: num,Q3: num] :
( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
<=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
tff(fact_3732_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,Nb: 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,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
tff(fact_3733_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,Nb: num] :
( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
<=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
tff(fact_3734_maxt__bound,axiom,
! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Ta)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).
% maxt_bound
tff(fact_3735_Suc__mod__eq__add3__mod,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = 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))),Mb)),Nb) ).
% Suc_mod_eq_add3_mod
tff(fact_3736_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(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_3737_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3738_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_3739_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3740_cong__exp__iff__simps_I9_J,axiom,
! [Mb: num,Q3: num,Nb: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).
% cong_exp_iff_simps(9)
tff(fact_3741_cong__exp__iff__simps_I9_J,axiom,
! [Mb: num,Q3: num,Nb: num] :
( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
<=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).
% cong_exp_iff_simps(9)
tff(fact_3742_cong__exp__iff__simps_I4_J,axiom,
! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).
% cong_exp_iff_simps(4)
tff(fact_3743_cong__exp__iff__simps_I4_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).
% cong_exp_iff_simps(4)
tff(fact_3744_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
=> ~ ! [D4: 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),D4)) ) ).
% mod_eqE
tff(fact_3745_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_3746_div__add1__eq,axiom,
! [A: int,B: int,C: int] : aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).
% div_add1_eq
tff(fact_3747_mod__Suc,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).
% mod_Suc
tff(fact_3748_mod__induct,axiom,
! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
( aa_nat_o(P,Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
=> ( aa_nat_o(P,N)
=> aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
=> aa_nat_o(P,Mb) ) ) ) ) ).
% mod_induct
tff(fact_3749_mod__less__divisor,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).
% mod_less_divisor
tff(fact_3750_mod__Suc__le__divisor,axiom,
! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).
% mod_Suc_le_divisor
tff(fact_3751_xor__num_Ocases,axiom,
! [Xb: product_prod_num_num] :
( ( Xb != product_Pair_num_num(one,one) )
=> ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N))
=> ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N))
=> ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),one)
=> ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))
=> ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))
=> ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),one)
=> ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))
=> ~ ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).
% xor_num.cases
tff(fact_3752_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] : Y != aa_num_num(bit0,X22)
=> ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).
% num.exhaust
tff(fact_3753_mod__eq__0D,axiom,
! [Mb: nat,D: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
=> ? [Q2: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).
% mod_eq_0D
tff(fact_3754_mod__if,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).
% mod_if
tff(fact_3755_mod__geq,axiom,
! [Mb: nat,Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).
% mod_geq
tff(fact_3756_le__mod__geq,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).
% le_mod_geq
tff(fact_3757_nat__mod__eq__iff,axiom,
! [Xb: nat,Nb: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
<=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).
% nat_mod_eq_iff
tff(fact_3758_infinite__Icc,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).
% infinite_Icc
tff(fact_3759_infinite__Icc,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B)) ) ).
% infinite_Icc
tff(fact_3760_finite__lists__length__le,axiom,
! [A3: set_complex,Nb: nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bd(A3),Nb))) ) ).
% finite_lists_length_le
tff(fact_3761_finite__lists__length__le,axiom,
! [A3: set_VEBT_VEBT,Nb: nat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_be(A3),Nb))) ) ).
% finite_lists_length_le
tff(fact_3762_finite__lists__length__le,axiom,
! [A3: set_o,Nb: nat] :
( finite_finite_o(A3)
=> finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bf(A3),Nb))) ) ).
% finite_lists_length_le
tff(fact_3763_finite__lists__length__le,axiom,
! [A3: set_int,Nb: nat] :
( aa_set_int_o(finite_finite_int,A3)
=> finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bg(A3),Nb))) ) ).
% finite_lists_length_le
tff(fact_3764_finite__lists__length__le,axiom,
! [A3: set_nat,Nb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bh(A3),Nb))) ) ).
% finite_lists_length_le
tff(fact_3765_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_3766_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3767_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_3768_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3769_cong__exp__iff__simps_I2_J,axiom,
! [Nb: num,Q3: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
tff(fact_3770_cong__exp__iff__simps_I2_J,axiom,
! [Nb: num,Q3: num] :
( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
<=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
tff(fact_3771_cong__exp__iff__simps_I1_J,axiom,
! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).
% cong_exp_iff_simps(1)
tff(fact_3772_cong__exp__iff__simps_I1_J,axiom,
! [Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).
% cong_exp_iff_simps(1)
tff(fact_3773_cong__exp__iff__simps_I8_J,axiom,
! [Mb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != 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,Q3))) ).
% cong_exp_iff_simps(8)
tff(fact_3774_cong__exp__iff__simps_I8_J,axiom,
! [Mb: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(8)
tff(fact_3775_cong__exp__iff__simps_I6_J,axiom,
! [Q3: num,Nb: 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,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(6)
tff(fact_3776_cong__exp__iff__simps_I6_J,axiom,
! [Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).
% cong_exp_iff_simps(6)
tff(fact_3777_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_3778_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(divide_divide_int(A),B))),aa_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_3779_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_3780_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(divide_divide_int(A),B)),B)),aa_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_3781_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_3782_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(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).
% mod_div_decomp
tff(fact_3783_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_3784_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(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).
% div_mult_mod_eq
tff(fact_3785_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_3786_mod__div__mult__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_3787_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_3788_mod__mult__div__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))) = A ).
% mod_mult_div_eq
tff(fact_3789_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_3790_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(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).
% mult_div_mod_eq
tff(fact_3791_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_3792_div__mult1__eq,axiom,
! [A: int,B: int,C: int] : aa_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(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C)) ).
% div_mult1_eq
tff(fact_3793_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_3794_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(divide_divide_int(A),B))) = aa_int_int(modulo_modulo_int(A),B) ).
% minus_mult_div_eq_mod
tff(fact_3795_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_3796_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(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B)) ).
% minus_mod_eq_mult_div
tff(fact_3797_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_3798_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(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_3799_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_3800_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(divide_divide_int(A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).
% minus_div_mult_eq_mod
tff(fact_3801_numeral__Bit1,axiom,
! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).
% numeral_Bit1
tff(fact_3802_numeral__Bit1,axiom,
! [Nb: num] : numera1916890842035813515d_enat(aa_num_num(bit1,Nb)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Nb),numera1916890842035813515d_enat(Nb)),one_on7984719198319812577d_enat) ).
% numeral_Bit1
tff(fact_3803_numeral__Bit1,axiom,
! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).
% numeral_Bit1
tff(fact_3804_numeral__Bit1,axiom,
! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).
% numeral_Bit1
tff(fact_3805_numeral__Bit1,axiom,
! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = 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,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).
% numeral_Bit1
tff(fact_3806_numeral__Bit1,axiom,
! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = 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,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).
% numeral_Bit1
tff(fact_3807_mod__le__divisor,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).
% mod_le_divisor
tff(fact_3808_div__less__mono,axiom,
! [A3: nat,B3: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),Nb) = zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ) ) ).
% div_less_mono
tff(fact_3809_eval__nat__numeral_I3_J,axiom,
! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).
% eval_nat_numeral(3)
tff(fact_3810_nat__mod__eq__lemma,axiom,
! [Xb: nat,Nb: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> ? [Q2: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).
% nat_mod_eq_lemma
tff(fact_3811_mod__eq__nat2E,axiom,
! [Mb: nat,Q3: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).
% mod_eq_nat2E
tff(fact_3812_mod__eq__nat1E,axiom,
! [Mb: nat,Q3: nat,Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ~ ! [S: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).
% mod_eq_nat1E
tff(fact_3813_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L != zero_zero_int )
=> ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int)) ) ) ).
% eucl_rel_int_dividesI
tff(fact_3814_mod__mult2__eq,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).
% mod_mult2_eq
tff(fact_3815_div__mod__decomp,axiom,
! [A3: nat,Nb: nat] : A3 = 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,A3),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),Nb)) ).
% div_mod_decomp
tff(fact_3816_modulo__nat__def,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)) ).
% modulo_nat_def
tff(fact_3817_numeral__code_I3_J,axiom,
! [Nb: num] :
aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
m2: rat,
m2:= aa_num_rat(numeral_numeral_rat,Nb),
aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).
% numeral_code(3)
tff(fact_3818_numeral__code_I3_J,axiom,
! [Nb: num] :
numera1916890842035813515d_enat(aa_num_num(bit1,Nb)) = $let(
m2: extended_enat,
m2:= numera1916890842035813515d_enat(Nb),
plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(m2,m2),one_on7984719198319812577d_enat) ) ).
% numeral_code(3)
tff(fact_3819_numeral__code_I3_J,axiom,
! [Nb: num] :
numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
m2: complex,
m2:= numera6690914467698888265omplex(Nb),
aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).
% numeral_code(3)
tff(fact_3820_numeral__code_I3_J,axiom,
! [Nb: num] :
aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
m2: real,
m2:= aa_num_real(numeral_numeral_real,Nb),
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).
% numeral_code(3)
tff(fact_3821_numeral__code_I3_J,axiom,
! [Nb: num] :
aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
m2: nat,
m2:= aa_num_nat(numeral_numeral_nat,Nb),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).
% numeral_code(3)
tff(fact_3822_numeral__code_I3_J,axiom,
! [Nb: num] :
aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
m2: int,
m2:= aa_num_int(numeral_numeral_int,Nb),
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).
% numeral_code(3)
tff(fact_3823_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))) = $let(
w: complex,
w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).
% power_numeral_odd
tff(fact_3824_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))) = $let(
w: real,
w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_3825_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))) = $let(
w: rat,
w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).
% power_numeral_odd
tff(fact_3826_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))) = $let(
w: nat,
w:= 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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_3827_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))) = $let(
w: int,
w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_3828_numeral__Bit1__div__2,axiom,
! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).
% numeral_Bit1_div_2
tff(fact_3829_numeral__Bit1__div__2,axiom,
! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).
% numeral_Bit1_div_2
tff(fact_3830_split__mod,axiom,
! [P: fun_nat_o,Mb: nat,Nb: nat] :
( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
<=> ( ( ( Nb = zero_zero_nat )
=> aa_nat_o(P,Mb) )
& ( ( Nb != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
=> ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I4)),J3) )
=> aa_nat_o(P,J3) ) ) ) ) ) ).
% split_mod
tff(fact_3831_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_3832_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_3833_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_3834_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_3835_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_3836_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_3837_Suc3__eq__add__3,axiom,
! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).
% Suc3_eq_add_3
tff(fact_3838_mint__bound,axiom,
! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Ta)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).
% mint_bound
tff(fact_3839_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_3840_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_3841_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_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(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3842_num_Osize_I6_J,axiom,
! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size(6)
tff(fact_3843_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,Nb: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,Nb)) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_3844_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va)),Vb) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_3845_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz)),Va) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_3846_Suc__times__mod__eq,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
=> ( 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,Mb),Nb))),Mb) = one_one_nat ) ) ).
% Suc_times_mod_eq
tff(fact_3847_Suc__div__eq__add3__div,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = 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))),Mb)),Nb) ).
% Suc_div_eq_add3_div
tff(fact_3848_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A: $o,B: $o] :
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),
$ite((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_3849_pred__bound__height,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d(Ta),Xb)),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,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).
% pred_bound_height
tff(fact_3850_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = one_one_nat ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_3851_succ__bound__height,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c(Ta),Xb)),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,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ) ).
% succ_bound_height
tff(fact_3852_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_3853_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_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(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(modulo_modulo_int(A),B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_3854_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_3855_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( aa_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(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_3856_div__exp__mod__exp__eq,axiom,
! [A: nat,Nb: nat,Mb: 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))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = 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,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% div_exp_mod_exp_eq
tff(fact_3857_div__exp__mod__exp__eq,axiom,
! [A: int,Nb: nat,Mb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(divide_divide_int(aa_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,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% div_exp_mod_exp_eq
tff(fact_3858_verit__le__mono__div,axiom,
! [A3: nat,B3: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),Nb)),
$ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ).
% verit_le_mono_div
tff(fact_3859_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
! [A: $o,B: $o,Va: nat] :
aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_3860_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve)),Vf) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_3861_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
! [Uu2: $o,B: $o] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Leaf((Uu2),(B))),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_3862_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd)),Ve) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_3863_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_3864_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_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(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(divide_divide_int(A),B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_3865_mult__exp__mod__exp__eq,axiom,
! [Mb: nat,Nb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( 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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_3866_mult__exp__mod__exp__eq,axiom,
! [Mb: nat,Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_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,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_3867_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mia: nat,Maa: 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(Mia),Maa)),Ux,Uy,Uz)) = one_one_nat ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_3868_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
<=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
& $ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
& aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
Q3 = zero_zero_int ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_3869_mod__double__modulus,axiom,
! [Mb: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).
% mod_double_modulus
tff(fact_3870_mod__double__modulus,axiom,
! [Mb: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( ( aa_int_int(modulo_modulo_int(Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(modulo_modulo_int(Xb),Mb) )
| ( aa_int_int(modulo_modulo_int(Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(Xb),Mb)),Mb) ) ) ) ) ).
% mod_double_modulus
tff(fact_3871_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_3872_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_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(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(modulo_modulo_int(A),B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_3873_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
! [A: $o,Uw2: $o] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((A),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_3874_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)),Vj) = one_one_nat ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_3875_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)),Vi) = one_one_nat ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_3876_unset__bit__Suc,axiom,
! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_3877_unset__bit__Suc,axiom,
! [Nb: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_3878_set__bit__Suc,axiom,
! [Nb: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_3879_set__bit__Suc,axiom,
! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_3880_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_3881_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_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(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(divide_divide_int(A),B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_3882_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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(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(bit0,one))))),one_one_nat)),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
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(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeLista,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeLista,h)),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summarya,h),
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,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summarya),h))),one_one_nat)),
$ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_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(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeLista,the_nat(pr)))))) ) )) ),
one_one_nat )) ) ) )) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_3883_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeLista,h)))),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeLista,h)),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summarya,h),
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,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summarya),h))),one_one_nat)),
$ite(sc = none_nat,one_one_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(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeLista,the_nat(sc)))))) ) )) )),
one_one_nat )) ) ) )) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_3884_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q3: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
=> aa_Pro4592068669816069595_int_o(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,Q3),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))),R2)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_3885_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_s_u_c_c(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,B4: $o] :
( ( Xb = vEBT_Leaf((Uu),(B4)) )
=> ( ( Xaa = zero_zero_nat )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B4)),zero_zero_nat)) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( Xb = vEBT_Leaf((Uv),(Uw)) )
=> ! [N: nat] :
( ( Xaa = aa_nat_nat(suc,N) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
=> ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
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(bit0,one))))),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,h)))),
$let(
maxlow: option_nat,
maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
$ite(
( ( maxlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList,h)),l)),
$let(
sc: option_nat,
sc:= vEBT_vebt_succ(Summary,h),
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,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary),h))),one_one_nat)),
$ite(sc = none_nat,one_one_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(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(sc)))))) ) )) )),
one_one_nat )) ) ) )) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_3886_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_p_r_e_d(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( ( Xaa = zero_zero_nat )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( Xb = vEBT_Leaf((A4),(Uw)) )
=> ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ! [Va2: nat] :
( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((B4),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
one_one_nat,
$let(
l: nat,
l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
$let(
minlow: option_nat,
minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
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(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
$ite(
( ( minlow != none_nat )
& aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList,h)),l)),
$let(
pr: option_nat,
pr:= vEBT_vebt_pred(Summary,h),
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,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary),h))),one_one_nat)),
$ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_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(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(pr)))))) ) )) ),
one_one_nat )) ) ) )) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_3887_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xb),Xaa) = Y )
=> ( ( ? [A4: $o,B4: $o] : Xb = 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))),
$ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
=> ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = 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] : Xb = 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)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$ite(
Xaa = Mi,
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
Xaa = Ma,
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
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))))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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_3888_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,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$ite(
Xb = Mia,
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
Xb = Maa,
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
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))))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),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(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),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_3889_mod__exhaust__less__4,axiom,
! [Mb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),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,Mb),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,Mb),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_3890_finite__Collect__le__nat,axiom,
! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bi,K))) ).
% finite_Collect_le_nat
tff(fact_3891_finite__Collect__less__nat,axiom,
! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,K))) ).
% finite_Collect_less_nat
tff(fact_3892_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = 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))),
$ite(Xaa = 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)),Xaa)) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ( 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,Uu,Uv,Uw),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = 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),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$ite(
Xaa = Mi,
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
Xaa = Ma,
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
one_one_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
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))))),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),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,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_3893_finite__interval__int4,axiom,
! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bk(A),B))) ).
% finite_interval_int4
tff(fact_3894_finite__interval__int2,axiom,
! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bl(A),B))) ).
% finite_interval_int2
tff(fact_3895_finite__interval__int3,axiom,
! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_bm(A),B))) ).
% finite_interval_int3
tff(fact_3896_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
=> ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).
% mod_pos_pos_trivial
tff(fact_3897_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
=> ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).
% mod_neg_neg_trivial
tff(fact_3898_zmod__numeral__Bit0,axiom,
! [V: num,W: num] : aa_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(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).
% zmod_numeral_Bit0
tff(fact_3899_zmod__numeral__Bit1,axiom,
! [V: num,W: num] : aa_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(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_3900_neg__mod__bound,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L)) ) ).
% neg_mod_bound
tff(fact_3901_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L) ) ).
% Euclidean_Division.pos_mod_bound
tff(fact_3902_zmod__eq__0__iff,axiom,
! [Mb: int,D: int] :
( ( aa_int_int(modulo_modulo_int(Mb),D) = zero_zero_int )
<=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).
% zmod_eq_0_iff
tff(fact_3903_zmod__eq__0D,axiom,
! [Mb: int,D: int] :
( ( aa_int_int(modulo_modulo_int(Mb),D) = zero_zero_int )
=> ? [Q2: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q2) ) ).
% zmod_eq_0D
tff(fact_3904_finite__maxlen,axiom,
! [M6: set_list_VEBT_VEBT] :
( finite3004134309566078307T_VEBT(M6)
=> ? [N: nat] :
! [X3: list_VEBT_VEBT] :
( member2936631157270082147T_VEBT(X3,M6)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X3)),N) ) ) ).
% finite_maxlen
tff(fact_3905_finite__maxlen,axiom,
! [M6: set_list_o] :
( finite_finite_list_o(M6)
=> ? [N: nat] :
! [X3: list_o] :
( member_list_o(X3,M6)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X3)),N) ) ) ).
% finite_maxlen
tff(fact_3906_finite__maxlen,axiom,
! [M6: set_list_nat] :
( finite8100373058378681591st_nat(M6)
=> ? [N: nat] :
! [X3: list_nat] :
( member_list_nat(X3,M6)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X3)),N) ) ) ).
% finite_maxlen
tff(fact_3907_finite__maxlen,axiom,
! [M6: set_list_int] :
( finite3922522038869484883st_int(M6)
=> ? [N: nat] :
! [X3: list_int] :
( member_list_int(X3,M6)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X3)),N) ) ) ).
% finite_maxlen
tff(fact_3908_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)) ) ).
% Euclidean_Division.pos_mod_sign
tff(fact_3909_neg__mod__sign,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int) ) ).
% neg_mod_sign
tff(fact_3910_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( aa_int_int(modulo_modulo_int(I),K) = I )
<=> ( ( K = zero_zero_int )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
& aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).
% zmod_trivial_iff
tff(fact_3911_pos__mod__conj,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ).
% pos_mod_conj
tff(fact_3912_neg__mod__conj,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ).
% neg_mod_conj
tff(fact_3913_zdiv__mono__strict,axiom,
! [A3: int,B3: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
=> ( ( aa_int_int(modulo_modulo_int(A3),Nb) = zero_zero_int )
=> ( ( aa_int_int(modulo_modulo_int(B3),Nb) = zero_zero_int )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A3),Nb)),aa_int_int(divide_divide_int(B3),Nb)) ) ) ) ) ).
% zdiv_mono_strict
tff(fact_3914_div__mod__decomp__int,axiom,
! [A3: int,Nb: int] : A3 = 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(divide_divide_int(A3),Nb)),Nb)),aa_int_int(modulo_modulo_int(A3),Nb)) ).
% div_mod_decomp_int
tff(fact_3915_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
=> ( aa_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_3916_mod__pos__geq,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
=> ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).
% mod_pos_geq
tff(fact_3917_int__mod__pos__eq,axiom,
! [A: int,B: int,Q3: int,R2: 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),Q3)),R2) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
=> ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).
% int_mod_pos_eq
tff(fact_3918_int__mod__neg__eq,axiom,
! [A: int,B: int,Q3: int,R2: 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),Q3)),R2) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
=> ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).
% int_mod_neg_eq
tff(fact_3919_split__zmod,axiom,
! [P: fun_int_o,Nb: int,K: int] :
( aa_int_o(P,aa_int_int(modulo_modulo_int(Nb),K))
<=> ( ( ( K = zero_zero_int )
=> aa_int_o(P,Nb) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ! [I4: int,J3: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
& ( Nb = 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) ) )
=> aa_int_o(P,J3) ) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
=> ! [I4: int,J3: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
& ( Nb = 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) ) )
=> aa_int_o(P,J3) ) ) ) ) ).
% split_zmod
tff(fact_3920_zmod__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_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(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).
% zmod_zmult2_eq
tff(fact_3921_verit__le__mono__div__int,axiom,
! [A3: int,B3: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A3),Nb)),
$ite(aa_int_int(modulo_modulo_int(B3),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(divide_divide_int(B3),Nb)) ) ) ).
% verit_le_mono_div_int
tff(fact_3922_split__pos__lemma,axiom,
! [K: int,P: fun_int_fun_int_o,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
<=> ! [I4: int,J3: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
& ( Nb = 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) ) )
=> aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).
% split_pos_lemma
tff(fact_3923_split__neg__lemma,axiom,
! [K: int,P: fun_int_fun_int_o,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
<=> ! [I4: int,J3: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
& ( Nb = 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) ) )
=> aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).
% split_neg_lemma
tff(fact_3924_finite__has__maximal2,axiom,
! [A3: set_set_nat,A: set_nat] :
( finite1152437895449049373et_nat(A3)
=> ( member_set_nat(A,A3)
=> ? [X4: set_nat] :
( member_set_nat(X4,A3)
& aa_set_nat_o(ord_less_eq_set_nat(A),X4)
& ! [Xa: set_nat] :
( member_set_nat(Xa,A3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3925_finite__has__maximal2,axiom,
! [A3: set_rat,A: rat] :
( aa_set_rat_o(finite_finite_rat,A3)
=> ( aa_set_rat_o(member_rat(A),A3)
=> ? [X4: rat] :
( aa_set_rat_o(member_rat(X4),A3)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X4)
& ! [Xa: rat] :
( aa_set_rat_o(member_rat(Xa),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3926_finite__has__maximal2,axiom,
! [A3: set_num,A: num] :
( aa_set_num_o(finite_finite_num,A3)
=> ( aa_set_num_o(member_num(A),A3)
=> ? [X4: num] :
( aa_set_num_o(member_num(X4),A3)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X4)
& ! [Xa: num] :
( aa_set_num_o(member_num(Xa),A3)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3927_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_set_nat_o(member_nat(A),A3)
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X4)
& ! [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3928_finite__has__maximal2,axiom,
! [A3: set_int,A: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(member_int(A),A3)
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),A3)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X4)
& ! [Xa: int] :
( aa_set_int_o(member_int(Xa),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3929_finite__has__maximal2,axiom,
! [A3: set_real,A: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( aa_set_real_o(member_real(A),A3)
=> ? [X4: real] :
( aa_set_real_o(member_real(X4),A3)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
& ! [Xa: real] :
( aa_set_real_o(member_real(Xa),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3930_finite__has__minimal2,axiom,
! [A3: set_set_nat,A: set_nat] :
( finite1152437895449049373et_nat(A3)
=> ( member_set_nat(A,A3)
=> ? [X4: set_nat] :
( member_set_nat(X4,A3)
& aa_set_nat_o(ord_less_eq_set_nat(X4),A)
& ! [Xa: set_nat] :
( member_set_nat(Xa,A3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3931_finite__has__minimal2,axiom,
! [A3: set_rat,A: rat] :
( aa_set_rat_o(finite_finite_rat,A3)
=> ( aa_set_rat_o(member_rat(A),A3)
=> ? [X4: rat] :
( aa_set_rat_o(member_rat(X4),A3)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),A)
& ! [Xa: rat] :
( aa_set_rat_o(member_rat(Xa),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3932_finite__has__minimal2,axiom,
! [A3: set_num,A: num] :
( aa_set_num_o(finite_finite_num,A3)
=> ( aa_set_num_o(member_num(A),A3)
=> ? [X4: num] :
( aa_set_num_o(member_num(X4),A3)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),A)
& ! [Xa: num] :
( aa_set_num_o(member_num(Xa),A3)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3933_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_set_nat_o(member_nat(A),A3)
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),A)
& ! [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3934_finite__has__minimal2,axiom,
! [A3: set_int,A: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(member_int(A),A3)
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),A3)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),A)
& ! [Xa: int] :
( aa_set_int_o(member_int(Xa),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3935_finite__has__minimal2,axiom,
! [A3: set_real,A: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( aa_set_real_o(member_real(A),A3)
=> ? [X4: real] :
( aa_set_real_o(member_real(X4),A3)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),A)
& ! [Xa: real] :
( aa_set_real_o(member_real(Xa),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3936_finite__psubset__induct,axiom,
! [A3: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ! [A6: set_nat] :
( aa_set_nat_o(finite_finite_nat,A6)
=> ( ! [B6: set_nat] :
( aa_set_nat_o(ord_less_set_nat(B6),A6)
=> aa_set_nat_o(P,B6) )
=> aa_set_nat_o(P,A6) ) )
=> aa_set_nat_o(P,A3) ) ) ).
% finite_psubset_induct
tff(fact_3937_finite__psubset__induct,axiom,
! [A3: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [A6: set_int] :
( aa_set_int_o(finite_finite_int,A6)
=> ( ! [B6: set_int] :
( aa_set_int_o(ord_less_set_int(B6),A6)
=> aa_set_int_o(P,B6) )
=> aa_set_int_o(P,A6) ) )
=> aa_set_int_o(P,A3) ) ) ).
% finite_psubset_induct
tff(fact_3938_finite__psubset__induct,axiom,
! [A3: set_complex,P: fun_set_complex_o] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [A6: set_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A6)
=> ( ! [B6: set_complex] :
( aa_set_complex_o(ord_less_set_complex(B6),A6)
=> aa_set_complex_o(P,B6) )
=> aa_set_complex_o(P,A6) ) )
=> aa_set_complex_o(P,A3) ) ) ).
% finite_psubset_induct
tff(fact_3939_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_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(modulo_modulo_int(B),A))) ) ) ).
% pos_zmod_mult_2
tff(fact_3940_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,
! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),Xb) = 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_3941_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_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(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_3942_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,Xb: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) = 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_3943_finite__has__minimal,axiom,
! [A3: set_set_nat] :
( finite1152437895449049373et_nat(A3)
=> ( ( A3 != bot_bot_set_set_nat )
=> ? [X4: set_nat] :
( member_set_nat(X4,A3)
& ! [Xa: set_nat] :
( member_set_nat(Xa,A3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3944_finite__has__minimal,axiom,
! [A3: set_rat] :
( aa_set_rat_o(finite_finite_rat,A3)
=> ( ( A3 != bot_bot_set_rat )
=> ? [X4: rat] :
( aa_set_rat_o(member_rat(X4),A3)
& ! [Xa: rat] :
( aa_set_rat_o(member_rat(Xa),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3945_finite__has__minimal,axiom,
! [A3: set_num] :
( aa_set_num_o(finite_finite_num,A3)
=> ( ( A3 != bot_bot_set_num )
=> ? [X4: num] :
( aa_set_num_o(member_num(X4),A3)
& ! [Xa: num] :
( aa_set_num_o(member_num(Xa),A3)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3946_finite__has__minimal,axiom,
! [A3: set_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( A3 != bot_bot_set_nat )
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
& ! [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3947_finite__has__minimal,axiom,
! [A3: set_int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( A3 != bot_bot_set_int )
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),A3)
& ! [Xa: int] :
( aa_set_int_o(member_int(Xa),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3948_finite__has__minimal,axiom,
! [A3: set_real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ( A3 != bot_bot_set_real )
=> ? [X4: real] :
( aa_set_real_o(member_real(X4),A3)
& ! [Xa: real] :
( aa_set_real_o(member_real(Xa),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3949_finite__has__maximal,axiom,
! [A3: set_set_nat] :
( finite1152437895449049373et_nat(A3)
=> ( ( A3 != bot_bot_set_set_nat )
=> ? [X4: set_nat] :
( member_set_nat(X4,A3)
& ! [Xa: set_nat] :
( member_set_nat(Xa,A3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3950_finite__has__maximal,axiom,
! [A3: set_rat] :
( aa_set_rat_o(finite_finite_rat,A3)
=> ( ( A3 != bot_bot_set_rat )
=> ? [X4: rat] :
( aa_set_rat_o(member_rat(X4),A3)
& ! [Xa: rat] :
( aa_set_rat_o(member_rat(Xa),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3951_finite__has__maximal,axiom,
! [A3: set_num] :
( aa_set_num_o(finite_finite_num,A3)
=> ( ( A3 != bot_bot_set_num )
=> ? [X4: num] :
( aa_set_num_o(member_num(X4),A3)
& ! [Xa: num] :
( aa_set_num_o(member_num(Xa),A3)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3952_finite__has__maximal,axiom,
! [A3: set_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( A3 != bot_bot_set_nat )
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
& ! [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3953_finite__has__maximal,axiom,
! [A3: set_int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( A3 != bot_bot_set_int )
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),A3)
& ! [Xa: int] :
( aa_set_int_o(member_int(Xa),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3954_finite__has__maximal,axiom,
! [A3: set_real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ( A3 != bot_bot_set_real )
=> ? [X4: real] :
( aa_set_real_o(member_real(X4),A3)
& ! [Xa: real] :
( aa_set_real_o(member_real(Xa),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa)
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3955_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,Xb: 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)),Xb) = 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_3956_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: $o,B: $o,Xb: nat] :
aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A),(B))),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$ite(Xb = 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_3957_member__bound__height,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r(Ta),Xb)),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,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).
% member_bound_height
tff(fact_3958_finite__nth__roots,axiom,
! [Nb: nat,C: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_bn(Nb),C))) ) ).
% finite_nth_roots
tff(fact_3959_finite__roots__unity,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
=> aa_set_real_o(finite_finite_real,collect_real(aTP_Lamm_bo(Nb))) ) ).
% finite_roots_unity
tff(fact_3960_finite__roots__unity,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
=> aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bp(Nb))) ) ).
% finite_roots_unity
tff(fact_3961_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xb),Xaa) = Y )
=> ( ( ? [A4: $o,B4: $o] : Xb = vEBT_Leaf((A4),(B4))
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(Xaa = 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,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S)
=> ( Y != one_one_nat ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary)
=> ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( 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)))))),
$let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
& ~ ( ( Xaa = Mi )
| ( Xaa = 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,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,h)))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),h),one_one_nat)),
one_one_nat ) ) )) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_3962_insert__bound__height,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Xb)),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,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))) ) ).
% insert_bound_height
tff(fact_3963_flip__bit__Suc,axiom,
! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_3964_flip__bit__Suc,axiom,
! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_3965_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,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb) = 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)))))),
$let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),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)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
& ~ ( ( Xb = Mia )
| ( Xb = Maa ) ) ),
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(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),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(TreeLista,h)))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summarya),h),one_one_nat)),
one_one_nat ) ) )) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_3966_divmod__algorithm__code_I8_J,axiom,
! [Mb: num,Nb: num] :
unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).
% divmod_algorithm_code(8)
tff(fact_3967_divmod__algorithm__code_I8_J,axiom,
! [Mb: num,Nb: num] :
unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).
% divmod_algorithm_code(8)
tff(fact_3968_divmod__algorithm__code_I8_J,axiom,
! [Mb: num,Nb: num] :
unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).
% divmod_algorithm_code(8)
tff(fact_3969_flip__bit__negative__int__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Nb),K)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% flip_bit_negative_int_iff
tff(fact_3970_divmod__algorithm__code_I2_J,axiom,
! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).
% divmod_algorithm_code(2)
tff(fact_3971_divmod__algorithm__code_I2_J,axiom,
! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).
% divmod_algorithm_code(2)
tff(fact_3972_divmod__algorithm__code_I3_J,axiom,
! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).
% divmod_algorithm_code(3)
tff(fact_3973_divmod__algorithm__code_I3_J,axiom,
! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = 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_3974_divmod__algorithm__code_I4_J,axiom,
! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).
% divmod_algorithm_code(4)
tff(fact_3975_divmod__algorithm__code_I4_J,axiom,
! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = 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_3976_divmod__algorithm__code_I7_J,axiom,
! [Mb: num,Nb: num] :
unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).
% divmod_algorithm_code(7)
tff(fact_3977_divmod__algorithm__code_I7_J,axiom,
! [Mb: num,Nb: num] :
unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).
% divmod_algorithm_code(7)
tff(fact_3978_divmod__algorithm__code_I7_J,axiom,
! [Mb: num,Nb: num] :
unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).
% divmod_algorithm_code(7)
tff(fact_3979_minNull__bound,axiom,
! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Ta)),one_one_nat) ).
% minNull_bound
tff(fact_3980_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($false,$false)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_3981_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,
! [Uv2: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($true,(Uv2))) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_3982_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,
! [Uu2: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf((Uu2),$true)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_3983_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,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero_nat,Ts,S2)),Xb) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_3984_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_3985_divmod__def,axiom,
! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).
% divmod_def
tff(fact_3986_divmod__def,axiom,
! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).
% divmod_def
tff(fact_3987_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,
! [Uw2: 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,Uw2,Ux,Uy)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_3988_divmod_H__nat__def,axiom,
! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).
% divmod'_nat_def
tff(fact_3989_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,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2)),Xb) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_3990_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: $o,B: $o,Xb: nat] :
aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A),(B))),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(Xb = 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_3991_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,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: 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)),TreeLista,Summarya)),Xb) = 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_3992_divmod__divmod__step,axiom,
! [Mb: num,Nb: num] :
unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).
% divmod_divmod_step
tff(fact_3993_divmod__divmod__step,axiom,
! [Mb: num,Nb: num] :
unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).
% divmod_divmod_step
tff(fact_3994_divmod__divmod__step,axiom,
! [Mb: num,Nb: num] :
unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).
% divmod_divmod_step
tff(fact_3995_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y )
=> ( ( ( Xb = vEBT_Leaf($false,$false) )
=> ( Y != one_one_nat ) )
=> ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
=> ( Y != one_one_nat ) )
=> ( ( ? [Uu: $o] : Xb = vEBT_Leaf((Uu),$true)
=> ( Y != one_one_nat ) )
=> ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
=> ( Y != one_one_nat ) )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
=> ( Y != one_one_nat ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_3996_insersimp,axiom,
! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_13)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).
% insersimp
tff(fact_3997_insertsimp,axiom,
! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).
% insertsimp
tff(fact_3998_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,
! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xb),Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite(Xaa = 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)),Xaa)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),Xaa)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xaa)) ) )
=> ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
=> ( ( 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)),TreeList,Summary),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( 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)))))),
$let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
& ~ ( ( Xaa = Mi )
| ( Xaa = 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,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),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,h)))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),h),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(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_3999_arcosh__1,axiom,
aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).
% arcosh_1
tff(fact_4000_product__nth,axiom,
! [Nb: nat,Xs: list_num,Ys: list_num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).
% product_nth
tff(fact_4001_product__nth,axiom,
! [Nb: nat,Xs: list_Code_integer,Ys: list_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
=> ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Nb) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).
% product_nth
tff(fact_4002_product__nth,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).
% product_nth
tff(fact_4003_product__nth,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
=> ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).
% product_nth
tff(fact_4004_product__nth,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).
% product_nth
tff(fact_4005_product__nth,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).
% product_nth
tff(fact_4006_product__nth,axiom,
! [Nb: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
=> ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).
% product_nth
tff(fact_4007_product__nth,axiom,
! [Nb: nat,Xs: list_o,Ys: list_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)))
=> ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Nb) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).
% product_nth
tff(fact_4008_product__nth,axiom,
! [Nb: nat,Xs: list_o,Ys: list_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)))
=> ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Nb) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).
% product_nth
tff(fact_4009_product__nth,axiom,
! [Nb: nat,Xs: list_o,Ys: list_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
=> ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).
% product_nth
tff(fact_4010_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_bq(I5),Xb)))
=> ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_bq(I5),Y)))
=> aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_br(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4011_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,Xb: fun_nat_complex,Y: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_bs(I5),Xb)))
=> ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_bs(I5),Y)))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_bt(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4012_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_bu(I5),Xb)))
=> ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_bu(I5),Y)))
=> aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_bv(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4013_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_bw(I5),Xb)))
=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_bw(I5),Y)))
=> aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_bx(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4014_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_by(I5),Xb)))
=> ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_by(I5),Y)))
=> aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_bz(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4015_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ca(I5),Xb)))
=> ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ca(I5),Y)))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_cb(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4016_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_cc(I5),Xb)))
=> ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_cc(I5),Y)))
=> aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_cd(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4017_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ce(I5),Xb)))
=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ce(I5),Y)))
=> aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_cf(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4018_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_cg(I5),Xb)))
=> ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_cg(I5),Y)))
=> aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ch(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4019_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ci(I5),Xb)))
=> ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ci(I5),Y)))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_cj(I5),Xb),Y))) ) ) ).
% prod.finite_Collect_op
tff(fact_4020_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ck(I5),Xb)))
=> ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ck(I5),Y)))
=> aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_cl(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4021_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,Xb: fun_nat_complex,Y: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_cm(I5),Xb)))
=> ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_cm(I5),Y)))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_cn(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4022_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_co(I5),Xb)))
=> ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_co(I5),Y)))
=> aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_cp(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4023_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_cq(I5),Xb)))
=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_cq(I5),Y)))
=> aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_cr(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4024_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),Xb)))
=> ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),Y)))
=> aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ct(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4025_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),Xb)))
=> ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),Y)))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_cv(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4026_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_cw(I5),Xb)))
=> ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_cw(I5),Y)))
=> aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_cx(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4027_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cy(I5),Xb)))
=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cy(I5),Y)))
=> aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_cz(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4028_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_da(I5),Xb)))
=> ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_da(I5),Y)))
=> aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_db(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4029_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dc(I5),Xb)))
=> ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dc(I5),Y)))
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dd(I5),Xb),Y))) ) ) ).
% sum.finite_Collect_op
tff(fact_4030_dbl__inc__simps_I3_J,axiom,
aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_4031_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_4032_dbl__inc__simps_I3_J,axiom,
aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_4033_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_4034_VEBT__internal_Oheight_Osimps_I1_J,axiom,
! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf((A),(B))) = zero_zero_nat ).
% VEBT_internal.height.simps(1)
tff(fact_4035_dbl__inc__simps_I2_J,axiom,
aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).
% dbl_inc_simps(2)
tff(fact_4036_dbl__inc__simps_I2_J,axiom,
aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).
% dbl_inc_simps(2)
tff(fact_4037_dbl__inc__simps_I2_J,axiom,
aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).
% dbl_inc_simps(2)
tff(fact_4038_dbl__inc__simps_I2_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).
% dbl_inc_simps(2)
tff(fact_4039_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_4040_dbl__inc__simps_I5_J,axiom,
! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).
% dbl_inc_simps(5)
tff(fact_4041_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_4042_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_4043_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).
% length_product
tff(fact_4044_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_4045_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_4046_length__product,axiom,
! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).
% length_product
tff(fact_4047_length__product,axiom,
! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).
% length_product
tff(fact_4048_length__product,axiom,
! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).
% length_product
tff(fact_4049_length__product,axiom,
! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).
% length_product
tff(fact_4050_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_4051_length__product,axiom,
! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).
% length_product
tff(fact_4052_dbl__inc__def,axiom,
! [Xb: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).
% dbl_inc_def
tff(fact_4053_dbl__inc__def,axiom,
! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).
% dbl_inc_def
tff(fact_4054_dbl__inc__def,axiom,
! [Xb: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xb) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).
% dbl_inc_def
tff(fact_4055_dbl__inc__def,axiom,
! [Xb: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).
% dbl_inc_def
tff(fact_4056_VEBT__internal_Oheight_Ocases,axiom,
! [Xb: vEBT_VEBT] :
( ! [A4: $o,B4: $o] : Xb != vEBT_Leaf((A4),(B4))
=> ~ ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb != vEBT_Node(Uu,Deg,TreeList,Summary) ) ).
% VEBT_internal.height.cases
tff(fact_4057_insert__simp__norm,axiom,
! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( ( Xb != Maa )
=> ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).
% insert_simp_norm
tff(fact_4058_insert__simp__excp,axiom,
! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
=> ( ( Xb != Maa )
=> ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).
% insert_simp_excp
tff(fact_4059_gcd__nat__induct,axiom,
! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
=> ( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N))
=> aa_nat_o(aa_nat_fun_nat_o(P,M2),N) ) )
=> aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ).
% gcd_nat_induct
tff(fact_4060_concat__bit__Suc,axiom,
! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Nb),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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(bit_concat_bit(Nb,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).
% concat_bit_Suc
tff(fact_4061_list__update__overwrite,axiom,
! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).
% list_update_overwrite
tff(fact_4062_max__bot2,axiom,
! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).
% max_bot2
tff(fact_4063_max__bot2,axiom,
! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).
% max_bot2
tff(fact_4064_max__bot2,axiom,
! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).
% max_bot2
tff(fact_4065_max__bot2,axiom,
! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),bot_bot_nat) = Xb ).
% max_bot2
tff(fact_4066_max__bot2,axiom,
! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),bot_bo4199563552545308370d_enat) = Xb ).
% max_bot2
tff(fact_4067_max__bot,axiom,
! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).
% max_bot
tff(fact_4068_max__bot,axiom,
! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).
% max_bot
tff(fact_4069_max__bot,axiom,
! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).
% max_bot
tff(fact_4070_max__bot,axiom,
! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xb) = Xb ).
% max_bot
tff(fact_4071_max__bot,axiom,
! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,bot_bo4199563552545308370d_enat),Xb) = Xb ).
% max_bot
tff(fact_4072_length__list__update,axiom,
! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).
% length_list_update
tff(fact_4073_length__list__update,axiom,
! [Xs: list_o,I: nat,Xb: $o] : size_size_list_o(list_update_o(Xs,I,(Xb))) = size_size_list_o(Xs) ).
% length_list_update
tff(fact_4074_length__list__update,axiom,
! [Xs: list_nat,I: nat,Xb: nat] : size_size_list_nat(list_update_nat(Xs,I,Xb)) = size_size_list_nat(Xs) ).
% length_list_update
tff(fact_4075_length__list__update,axiom,
! [Xs: list_int,I: nat,Xb: int] : size_size_list_int(list_update_int(Xs,I,Xb)) = size_size_list_int(Xs) ).
% length_list_update
tff(fact_4076_max__Suc__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).
% max_Suc_Suc
tff(fact_4077_max__0R,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).
% max_0R
tff(fact_4078_max__0L,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).
% max_0L
tff(fact_4079_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_4080_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_4081_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_4082_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_4083_list__update__id,axiom,
! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).
% list_update_id
tff(fact_4084_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_4085_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_4086_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_int,Xb: int] :
( ( I != J )
=> ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).
% nth_list_update_neq
tff(fact_4087_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,Xb: nat] :
( ( I != J )
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).
% nth_list_update_neq
tff(fact_4088_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
( ( I != J )
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).
% nth_list_update_neq
tff(fact_4089_max__number__of_I1_J,axiom,
! [U: num,V: num] :
aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).
% max_number_of(1)
tff(fact_4090_max__number__of_I1_J,axiom,
! [U: num,V: num] :
aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).
% max_number_of(1)
tff(fact_4091_max__number__of_I1_J,axiom,
! [U: num,V: num] :
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)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).
% max_number_of(1)
tff(fact_4092_max__number__of_I1_J,axiom,
! [U: num,V: num] :
aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).
% max_number_of(1)
tff(fact_4093_max__number__of_I1_J,axiom,
! [U: num,V: num] :
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).
% max_number_of(1)
tff(fact_4094_max__0__1_I3_J,axiom,
! [Xb: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).
% max_0_1(3)
tff(fact_4095_max__0__1_I3_J,axiom,
! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).
% max_0_1(3)
tff(fact_4096_max__0__1_I3_J,axiom,
! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).
% max_0_1(3)
tff(fact_4097_max__0__1_I3_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).
% max_0_1(3)
tff(fact_4098_max__0__1_I3_J,axiom,
! [Xb: num] : aa_int_int(ord_max_int(zero_zero_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).
% max_0_1(3)
tff(fact_4099_max__0__1_I4_J,axiom,
! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).
% max_0_1(4)
tff(fact_4100_max__0__1_I4_J,axiom,
! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).
% max_0_1(4)
tff(fact_4101_max__0__1_I4_J,axiom,
! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xb)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xb) ).
% max_0_1(4)
tff(fact_4102_max__0__1_I4_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).
% max_0_1(4)
tff(fact_4103_max__0__1_I4_J,axiom,
! [Xb: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xb)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xb) ).
% max_0_1(4)
tff(fact_4104_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_4105_max__0__1_I1_J,axiom,
aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).
% max_0_1(1)
tff(fact_4106_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_4107_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_4108_max__0__1_I1_J,axiom,
aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).
% max_0_1(1)
tff(fact_4109_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_4110_max__0__1_I2_J,axiom,
aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).
% max_0_1(2)
tff(fact_4111_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_4112_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_4113_max__0__1_I2_J,axiom,
aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).
% max_0_1(2)
tff(fact_4114_max__0__1_I5_J,axiom,
! [Xb: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).
% max_0_1(5)
tff(fact_4115_max__0__1_I5_J,axiom,
! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).
% max_0_1(5)
tff(fact_4116_max__0__1_I5_J,axiom,
! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).
% max_0_1(5)
tff(fact_4117_max__0__1_I5_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).
% max_0_1(5)
tff(fact_4118_max__0__1_I5_J,axiom,
! [Xb: num] : aa_int_int(ord_max_int(one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).
% max_0_1(5)
tff(fact_4119_max__0__1_I6_J,axiom,
! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).
% max_0_1(6)
tff(fact_4120_max__0__1_I6_J,axiom,
! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).
% max_0_1(6)
tff(fact_4121_max__0__1_I6_J,axiom,
! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) = aa_num_real(numeral_numeral_real,Xb) ).
% max_0_1(6)
tff(fact_4122_max__0__1_I6_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).
% max_0_1(6)
tff(fact_4123_max__0__1_I6_J,axiom,
! [Xb: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xb)),one_one_int) = aa_num_int(numeral_numeral_int,Xb) ).
% max_0_1(6)
tff(fact_4124_list__update__beyond,axiom,
! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
=> ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs ) ) ).
% list_update_beyond
tff(fact_4125_list__update__beyond,axiom,
! [Xs: list_o,I: nat,Xb: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
=> ( list_update_o(Xs,I,(Xb)) = Xs ) ) ).
% list_update_beyond
tff(fact_4126_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
=> ( list_update_nat(Xs,I,Xb) = Xs ) ) ).
% list_update_beyond
tff(fact_4127_list__update__beyond,axiom,
! [Xs: list_int,I: nat,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I)
=> ( list_update_int(Xs,I,Xb) = Xs ) ) ).
% list_update_beyond
tff(fact_4128_concat__bit__negative__iff,axiom,
! [Nb: nat,K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Nb,K),L)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).
% concat_bit_negative_iff
tff(fact_4129_nth__list__update__eq,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I) = Xb ) ) ).
% nth_list_update_eq
tff(fact_4130_nth__list__update__eq,axiom,
! [I: nat,Xs: list_o,Xb: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
=> ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),I)
<=> (Xb) ) ) ).
% nth_list_update_eq
tff(fact_4131_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),I) = Xb ) ) ).
% nth_list_update_eq
tff(fact_4132_nth__list__update__eq,axiom,
! [I: nat,Xs: list_int,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
=> ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),I) = Xb ) ) ).
% nth_list_update_eq
tff(fact_4133_set__swap,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4134_set__swap,axiom,
! [I: nat,Xs: list_o,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
=> ( set_o2(list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).
% set_swap
tff(fact_4135_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4136_set__swap,axiom,
! [I: nat,Xs: list_int,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4137_list__update__swap,axiom,
! [I: nat,I6: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X5: vEBT_VEBT] :
( ( I != I6 )
=> ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I6,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I6,X5),I,Xb) ) ) ).
% list_update_swap
tff(fact_4138_max__absorb2,axiom,
! [Xb: extended_enat,Y: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Xb),Y)
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y) = Y ) ) ).
% max_absorb2
tff(fact_4139_max__absorb2,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> ( ord_max_set_nat(Xb,Y) = Y ) ) ).
% max_absorb2
tff(fact_4140_max__absorb2,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> ( aa_rat_rat(ord_max_rat(Xb),Y) = Y ) ) ).
% max_absorb2
tff(fact_4141_max__absorb2,axiom,
! [Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y) = Y ) ) ).
% max_absorb2
tff(fact_4142_max__absorb2,axiom,
! [Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Y ) ) ).
% max_absorb2
tff(fact_4143_max__absorb2,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> ( aa_int_int(ord_max_int(Xb),Y) = Y ) ) ).
% max_absorb2
tff(fact_4144_max__absorb2,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y) = Y ) ) ).
% max_absorb2
tff(fact_4145_max__absorb1,axiom,
! [Y: extended_enat,Xb: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y),Xb)
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y) = Xb ) ) ).
% max_absorb1
tff(fact_4146_max__absorb1,axiom,
! [Y: set_nat,Xb: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Y),Xb)
=> ( ord_max_set_nat(Xb,Y) = Xb ) ) ).
% max_absorb1
tff(fact_4147_max__absorb1,axiom,
! [Y: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> ( aa_rat_rat(ord_max_rat(Xb),Y) = Xb ) ) ).
% max_absorb1
tff(fact_4148_max__absorb1,axiom,
! [Y: num,Xb: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y) = Xb ) ) ).
% max_absorb1
tff(fact_4149_max__absorb1,axiom,
! [Y: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Xb ) ) ).
% max_absorb1
tff(fact_4150_max__absorb1,axiom,
! [Y: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
=> ( aa_int_int(ord_max_int(Xb),Y) = Xb ) ) ).
% max_absorb1
tff(fact_4151_max__absorb1,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y) = Xb ) ) ).
% max_absorb1
tff(fact_4152_max__def,axiom,
! [A: extended_enat,B: extended_enat] :
aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B),B,A) ).
% max_def
tff(fact_4153_max__def,axiom,
! [A: set_nat,B: set_nat] :
ord_max_set_nat(A,B) = $ite(aa_set_nat_o(ord_less_eq_set_nat(A),B),B,A) ).
% max_def
tff(fact_4154_max__def,axiom,
! [A: rat,B: rat] :
aa_rat_rat(ord_max_rat(A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ).
% max_def
tff(fact_4155_max__def,axiom,
! [A: num,B: num] :
aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ).
% max_def
tff(fact_4156_max__def,axiom,
! [A: nat,B: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).
% max_def
tff(fact_4157_max__def,axiom,
! [A: int,B: int] :
aa_int_int(ord_max_int(A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).
% max_def
tff(fact_4158_max__def,axiom,
! [A: real,B: real] :
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B),B,A) ).
% max_def
tff(fact_4159_max__add__distrib__right,axiom,
! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)) ).
% max_add_distrib_right
tff(fact_4160_max__add__distrib__right,axiom,
! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_rat_rat(plus_plus_rat(Xb),Z)) ).
% max_add_distrib_right
tff(fact_4161_max__add__distrib__right,axiom,
! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),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,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).
% max_add_distrib_right
tff(fact_4162_max__add__distrib__right,axiom,
! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).
% max_add_distrib_right
tff(fact_4163_max__add__distrib__left,axiom,
! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).
% max_add_distrib_left
tff(fact_4164_max__add__distrib__left,axiom,
! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xb),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).
% max_add_distrib_left
tff(fact_4165_max__add__distrib__left,axiom,
! [Xb: 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,Xb),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).
% max_add_distrib_left
tff(fact_4166_max__add__distrib__left,axiom,
! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(Xb),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).
% max_add_distrib_left
tff(fact_4167_max__diff__distrib__left,axiom,
! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).
% max_diff_distrib_left
tff(fact_4168_max__diff__distrib__left,axiom,
! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xb),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xb),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).
% max_diff_distrib_left
tff(fact_4169_max__diff__distrib__left,axiom,
! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ord_max_int(Xb),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).
% max_diff_distrib_left
tff(fact_4170_nat__add__max__right,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ).
% nat_add_max_right
tff(fact_4171_nat__add__max__left,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q3)) ).
% nat_add_max_left
tff(fact_4172_nat__mult__max__right,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).
% nat_mult_max_right
tff(fact_4173_nat__mult__max__left,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).
% nat_mult_max_left
tff(fact_4174_max__def__raw,axiom,
! [X3: extended_enat,Xa: extended_enat] :
aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X3),Xa) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4175_max__def__raw,axiom,
! [X3: set_nat,Xa: set_nat] :
ord_max_set_nat(X3,Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4176_max__def__raw,axiom,
! [X3: rat,Xa: rat] :
aa_rat_rat(ord_max_rat(X3),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4177_max__def__raw,axiom,
! [X3: num,Xa: num] :
aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4178_max__def__raw,axiom,
! [X3: nat,Xa: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4179_max__def__raw,axiom,
! [X3: int,Xa: int] :
aa_int_int(ord_max_int(X3),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4180_max__def__raw,axiom,
! [X3: real,Xa: real] :
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X3),Xa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa),Xa,X3) ).
% max_def_raw
tff(fact_4181_concat__bit__assoc,axiom,
! [Nb: nat,K: int,Mb: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Nb,K),aa_int_int(bit_concat_bit(Mb,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb),aa_int_int(bit_concat_bit(Nb,K),L)),R2) ).
% concat_bit_assoc
tff(fact_4182_nat__minus__add__max,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).
% nat_minus_add_max
tff(fact_4183_set__update__subsetI,axiom,
! [Xs: list_P6011104703257516679at_nat,A3: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat,I: nat] :
( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),A3)
=> ( member8440522571783428010at_nat(Xb,A3)
=> ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,Xb)),A3) ) ) ).
% set_update_subsetI
tff(fact_4184_set__update__subsetI,axiom,
! [Xs: list_complex,A3: set_complex,Xb: complex,I: nat] :
( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,Xb))),A3) ) ) ).
% set_update_subsetI
tff(fact_4185_set__update__subsetI,axiom,
! [Xs: list_real,A3: set_real,Xb: real,I: nat] :
( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A3)
=> ( aa_set_real_o(member_real(Xb),A3)
=> aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb))),A3) ) ) ).
% set_update_subsetI
tff(fact_4186_set__update__subsetI,axiom,
! [Xs: list_int,A3: set_int,Xb: int,I: nat] :
( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A3)
=> ( aa_set_int_o(member_int(Xb),A3)
=> aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),A3) ) ) ).
% set_update_subsetI
tff(fact_4187_set__update__subsetI,axiom,
! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,Xb: vEBT_VEBT,I: nat] :
( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb))),A3) ) ) ).
% set_update_subsetI
tff(fact_4188_set__update__subsetI,axiom,
! [Xs: list_nat,A3: set_nat,Xb: nat,I: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A3)
=> ( aa_set_nat_o(member_nat(Xb),A3)
=> aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xb))),A3) ) ) ).
% set_update_subsetI
tff(fact_4189_set__update__memI,axiom,
! [Nb: nat,Xs: list_P6011104703257516679at_nat,Xb: product_prod_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s5460976970255530739at_nat(Xs))
=> member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,Nb,Xb))) ) ).
% set_update_memI
tff(fact_4190_set__update__memI,axiom,
! [Nb: nat,Xs: list_complex,Xb: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
=> aa_set_complex_o(member_complex(Xb),set_complex2(list_update_complex(Xs,Nb,Xb))) ) ).
% set_update_memI
tff(fact_4191_set__update__memI,axiom,
! [Nb: nat,Xs: list_real,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
=> aa_set_real_o(member_real(Xb),set_real2(list_update_real(Xs,Nb,Xb))) ) ).
% set_update_memI
tff(fact_4192_set__update__memI,axiom,
! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).
% set_update_memI
tff(fact_4193_set__update__memI,axiom,
! [Nb: nat,Xs: list_o,Xb: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
=> member_o((Xb),set_o2(list_update_o(Xs,Nb,(Xb)))) ) ).
% set_update_memI
tff(fact_4194_set__update__memI,axiom,
! [Nb: nat,Xs: list_nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
=> aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).
% set_update_memI
tff(fact_4195_set__update__memI,axiom,
! [Nb: nat,Xs: list_int,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
=> aa_set_int_o(member_int(Xb),set_int2(list_update_int(Xs,Nb,Xb))) ) ).
% set_update_memI
tff(fact_4196_nth__list__update,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).
% nth_list_update
tff(fact_4197_nth__list__update,axiom,
! [I: nat,Xs: list_o,Xb: $o,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
=> ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),J)
<=> $ite(I = J,(Xb),aa_nat_o(nth_o(Xs),J)) ) ) ).
% nth_list_update
tff(fact_4198_nth__list__update,axiom,
! [I: nat,Xs: list_nat,Xb: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).
% nth_list_update
tff(fact_4199_nth__list__update,axiom,
! [I: nat,Xs: list_int,Xb: int,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
=> ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).
% nth_list_update
tff(fact_4200_list__update__same__conv,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
=> ( ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs )
<=> ( nth_VEBT_VEBT(Xs,I) = Xb ) ) ) ).
% list_update_same_conv
tff(fact_4201_list__update__same__conv,axiom,
! [I: nat,Xs: list_o,Xb: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
=> ( ( list_update_o(Xs,I,(Xb)) = Xs )
<=> ( aa_nat_o(nth_o(Xs),I)
<=> (Xb) ) ) ) ).
% list_update_same_conv
tff(fact_4202_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
=> ( ( list_update_nat(Xs,I,Xb) = Xs )
<=> ( aa_nat_nat(nth_nat(Xs),I) = Xb ) ) ) ).
% list_update_same_conv
tff(fact_4203_list__update__same__conv,axiom,
! [I: nat,Xs: list_int,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
=> ( ( list_update_int(Xs,I,Xb) = Xs )
<=> ( aa_nat_int(nth_int(Xs),I) = Xb ) ) ) ).
% list_update_same_conv
tff(fact_4204_prod__decode__aux_Ocases,axiom,
! [Xb: product_prod_nat_nat] :
~ ! [K3: nat,M2: nat] : Xb != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K3),M2) ).
% prod_decode_aux.cases
tff(fact_4205_list__decode_Ocases,axiom,
! [Xb: nat] :
( ( Xb != zero_zero_nat )
=> ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).
% list_decode.cases
tff(fact_4206_Euclid__induct,axiom,
! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,A4),B4)
<=> aa_nat_o(aa_nat_fun_nat_o(P,B4),A4) )
=> ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),zero_zero_nat)
=> ( ! [A4: nat,B4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,A4),B4)
=> aa_nat_o(aa_nat_fun_nat_o(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)) )
=> aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).
% Euclid_induct
tff(fact_4207_vebt__insert_Osimps_I5_J,axiom,
! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),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)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
& ~ ( ( Xb = Mia )
| ( Xb = Maa ) ) ),
vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),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)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).
% vebt_insert.simps(5)
tff(fact_4208_vebt__insert_Oelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_insert(Xb,Xaa) = Y )
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( Y != $ite(
Xaa = zero_zero_nat,
vEBT_Leaf($true,(B4)),
$ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
=> ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
=> ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ) )
=> ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
=> ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( Y != $let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
& ~ ( ( Xaa = Mi )
| ( Xaa = Ma ) ) ),
vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
tff(fact_4209_vebt__insert_Opelims,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_insert(Xb,Xaa) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = $ite(
Xaa = zero_zero_nat,
vEBT_Leaf($true,(B4)),
$ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
=> ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),Xaa)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
=> ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xaa)) ) )
=> ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
=> ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
=> ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
=> ( ( Y = $let(
xn: nat,
xn:=
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
$let(
h: nat,
h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
$ite(
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
& ~ ( ( Xaa = Mi )
| ( Xaa = Ma ) ) ),
vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
$ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
tff(fact_4210_max_Oabsorb3,axiom,
! [B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).
% max.absorb3
tff(fact_4211_max_Oabsorb3,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).
% max.absorb3
tff(fact_4212_max_Oabsorb3,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).
% max.absorb3
tff(fact_4213_max_Oabsorb3,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).
% max.absorb3
tff(fact_4214_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).
% max.absorb3
tff(fact_4215_max_Oabsorb3,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_int(ord_max_int(A),B) = A ) ) ).
% max.absorb3
tff(fact_4216_max_Oabsorb4,axiom,
! [A: extended_enat,B: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).
% max.absorb4
tff(fact_4217_max_Oabsorb4,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).
% max.absorb4
tff(fact_4218_max_Oabsorb4,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).
% max.absorb4
tff(fact_4219_max_Oabsorb4,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).
% max.absorb4
tff(fact_4220_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).
% max.absorb4
tff(fact_4221_max_Oabsorb4,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_int(ord_max_int(A),B) = B ) ) ).
% max.absorb4
tff(fact_4222_max__less__iff__conj,axiom,
! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y)),Z)
<=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Xb),Z)
& aa_Extended_enat_o(ord_le72135733267957522d_enat(Y),Z) ) ) ).
% max_less_iff_conj
tff(fact_4223_max__less__iff__conj,axiom,
! [Xb: real,Y: real,Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).
% max_less_iff_conj
tff(fact_4224_max__less__iff__conj,axiom,
! [Xb: rat,Y: rat,Z: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xb),Y)),Z)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).
% max_less_iff_conj
tff(fact_4225_max__less__iff__conj,axiom,
! [Xb: num,Y: num,Z: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y)),Z)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z)
& aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z) ) ) ).
% max_less_iff_conj
tff(fact_4226_max__less__iff__conj,axiom,
! [Xb: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).
% max_less_iff_conj
tff(fact_4227_max__less__iff__conj,axiom,
! [Xb: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Xb),Y)),Z)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).
% max_less_iff_conj
tff(fact_4228_max_Obounded__iff,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
<=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
& aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A) ) ) ).
% max.bounded_iff
tff(fact_4229_max_Obounded__iff,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).
% max.bounded_iff
tff(fact_4230_max_Obounded__iff,axiom,
! [B: num,C: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
& aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).
% max.bounded_iff
tff(fact_4231_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).
% max.bounded_iff
tff(fact_4232_max_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).
% max.bounded_iff
tff(fact_4233_max_Obounded__iff,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A) ) ) ).
% max.bounded_iff
tff(fact_4234_max_Oabsorb2,axiom,
! [A: extended_enat,B: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).
% max.absorb2
tff(fact_4235_max_Oabsorb2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).
% max.absorb2
tff(fact_4236_max_Oabsorb2,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).
% max.absorb2
tff(fact_4237_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).
% max.absorb2
tff(fact_4238_max_Oabsorb2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_int(ord_max_int(A),B) = B ) ) ).
% max.absorb2
tff(fact_4239_max_Oabsorb2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).
% max.absorb2
tff(fact_4240_max_Oabsorb1,axiom,
! [B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).
% max.absorb1
tff(fact_4241_max_Oabsorb1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).
% max.absorb1
tff(fact_4242_max_Oabsorb1,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).
% max.absorb1
tff(fact_4243_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).
% max.absorb1
tff(fact_4244_max_Oabsorb1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_int(ord_max_int(A),B) = A ) ) ).
% max.absorb1
tff(fact_4245_max_Oabsorb1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).
% max.absorb1
tff(fact_4246_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q3) = Q3 ).
% max_enat_simps(3)
tff(fact_4247_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q3),zero_z5237406670263579293d_enat) = Q3 ).
% max_enat_simps(2)
tff(fact_4248_max_OcoboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),B)
=> aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).
% max.coboundedI2
tff(fact_4249_max_OcoboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).
% max.coboundedI2
tff(fact_4250_max_OcoboundedI2,axiom,
! [C: num,B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).
% max.coboundedI2
tff(fact_4251_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).
% max.coboundedI2
tff(fact_4252_max_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(ord_max_int(A),B)) ) ).
% max.coboundedI2
tff(fact_4253_max_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).
% max.coboundedI2
tff(fact_4254_max_OcoboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
=> aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).
% max.coboundedI1
tff(fact_4255_max_OcoboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).
% max.coboundedI1
tff(fact_4256_max_OcoboundedI1,axiom,
! [C: num,A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).
% max.coboundedI1
tff(fact_4257_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).
% max.coboundedI1
tff(fact_4258_max_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(ord_max_int(A),B)) ) ).
% max.coboundedI1
tff(fact_4259_max_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).
% max.coboundedI1
tff(fact_4260_max_Oabsorb__iff2,axiom,
! [A: extended_enat,B: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
<=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_4261_max_Oabsorb__iff2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
<=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_4262_max_Oabsorb__iff2,axiom,
! [A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(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_4263_max_Oabsorb__iff2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4264_max_Oabsorb__iff2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_int(ord_max_int(A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_4265_max_Oabsorb__iff2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
<=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_4266_max_Oabsorb__iff1,axiom,
! [B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
<=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_4267_max_Oabsorb__iff1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
<=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_4268_max_Oabsorb__iff1,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(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_4269_max_Oabsorb__iff1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4270_max_Oabsorb__iff1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( aa_int_int(ord_max_int(A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_4271_max_Oabsorb__iff1,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
<=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_4272_le__max__iff__disj,axiom,
! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y))
<=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Xb)
| aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y) ) ) ).
% le_max_iff_disj
tff(fact_4273_le__max__iff__disj,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y))
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) ) ) ).
% le_max_iff_disj
tff(fact_4274_le__max__iff__disj,axiom,
! [Z: num,Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
| aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) ) ) ).
% le_max_iff_disj
tff(fact_4275_le__max__iff__disj,axiom,
! [Z: nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).
% le_max_iff_disj
tff(fact_4276_le__max__iff__disj,axiom,
! [Z: int,Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(ord_max_int(Xb),Y))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).
% le_max_iff_disj
tff(fact_4277_le__max__iff__disj,axiom,
! [Z: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y))
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Xb)
| aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Y) ) ) ).
% le_max_iff_disj
tff(fact_4278_max_Ocobounded2,axiom,
! [B: extended_enat,A: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ).
% max.cobounded2
tff(fact_4279_max_Ocobounded2,axiom,
! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B)) ).
% max.cobounded2
tff(fact_4280_max_Ocobounded2,axiom,
! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).
% max.cobounded2
tff(fact_4281_max_Ocobounded2,axiom,
! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).
% max.cobounded2
tff(fact_4282_max_Ocobounded2,axiom,
! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(ord_max_int(A),B)) ).
% max.cobounded2
tff(fact_4283_max_Ocobounded2,axiom,
! [B: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ).
% max.cobounded2
tff(fact_4284_max_Ocobounded1,axiom,
! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ).
% max.cobounded1
tff(fact_4285_max_Ocobounded1,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B)) ).
% max.cobounded1
tff(fact_4286_max_Ocobounded1,axiom,
! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).
% max.cobounded1
tff(fact_4287_max_Ocobounded1,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).
% max.cobounded1
tff(fact_4288_max_Ocobounded1,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(ord_max_int(A),B)) ).
% max.cobounded1
tff(fact_4289_max_Ocobounded1,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ).
% max.cobounded1
tff(fact_4290_max_Oorder__iff,axiom,
! [B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
<=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).
% max.order_iff
tff(fact_4291_max_Oorder__iff,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
<=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).
% max.order_iff
tff(fact_4292_max_Oorder__iff,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(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_4293_max_Oorder__iff,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4294_max_Oorder__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( A = aa_int_int(ord_max_int(A),B) ) ) ).
% max.order_iff
tff(fact_4295_max_Oorder__iff,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
<=> ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) ) ) ).
% max.order_iff
tff(fact_4296_max_OboundedI,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
=> aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A) ) ) ).
% max.boundedI
tff(fact_4297_max_OboundedI,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A) ) ) ).
% max.boundedI
tff(fact_4298_max_OboundedI,axiom,
! [B: num,A: num,C: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A) ) ) ).
% max.boundedI
tff(fact_4299_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).
% max.boundedI
tff(fact_4300_max_OboundedI,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A) ) ) ).
% max.boundedI
tff(fact_4301_max_OboundedI,axiom,
! [B: real,A: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A) ) ) ).
% max.boundedI
tff(fact_4302_max_OboundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
=> ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
=> ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A) ) ) ).
% max.boundedE
tff(fact_4303_max_OboundedE,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)
=> ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).
% max.boundedE
tff(fact_4304_max_OboundedE,axiom,
! [B: num,C: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
=> ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).
% max.boundedE
tff(fact_4305_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).
% max.boundedE
tff(fact_4306_max_OboundedE,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A)
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).
% max.boundedE
tff(fact_4307_max_OboundedE,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
=> ~ ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A) ) ) ).
% max.boundedE
tff(fact_4308_max_OorderI,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
=> aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A) ) ).
% max.orderI
tff(fact_4309_max_OorderI,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(ord_max_rat(A),B) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).
% max.orderI
tff(fact_4310_max_OorderI,axiom,
! [A: num,B: num] :
( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).
% max.orderI
tff(fact_4311_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).
% max.orderI
tff(fact_4312_max_OorderI,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(ord_max_int(A),B) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% max.orderI
tff(fact_4313_max_OorderI,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).
% max.orderI
tff(fact_4314_max_OorderE,axiom,
! [B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).
% max.orderE
tff(fact_4315_max_OorderE,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).
% max.orderE
tff(fact_4316_max_OorderE,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).
% max.orderE
tff(fact_4317_max_OorderE,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).
% max.orderE
tff(fact_4318_max_OorderE,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( A = aa_int_int(ord_max_int(A),B) ) ) ).
% max.orderE
tff(fact_4319_max_OorderE,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
=> ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) ) ) ).
% max.orderE
tff(fact_4320_max_Omono,axiom,
! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(D),B)
=> aa_Extended_enat_o(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_4321_max_Omono,axiom,
! [C: rat,A: rat,D: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D)),aa_rat_rat(ord_max_rat(A),B)) ) ) ).
% max.mono
tff(fact_4322_max_Omono,axiom,
! [C: num,A: num,D: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
=> aa_num_o(aa_num_fun_num_o(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_4323_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_4324_max_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(C),D)),aa_int_int(ord_max_int(A),B)) ) ) ).
% max.mono
tff(fact_4325_max_Omono,axiom,
! [C: real,A: real,D: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,C),D)),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ) ).
% max.mono
tff(fact_4326_max_Ostrict__coboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),B)
=> aa_Extended_enat_o(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).
% max.strict_coboundedI2
tff(fact_4327_max_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).
% max.strict_coboundedI2
tff(fact_4328_max_Ostrict__coboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).
% max.strict_coboundedI2
tff(fact_4329_max_Ostrict__coboundedI2,axiom,
! [C: num,B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).
% max.strict_coboundedI2
tff(fact_4330_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).
% max.strict_coboundedI2
tff(fact_4331_max_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).
% max.strict_coboundedI2
tff(fact_4332_max_Ostrict__coboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A)
=> aa_Extended_enat_o(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).
% max.strict_coboundedI1
tff(fact_4333_max_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).
% max.strict_coboundedI1
tff(fact_4334_max_Ostrict__coboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).
% max.strict_coboundedI1
tff(fact_4335_max_Ostrict__coboundedI1,axiom,
! [C: num,A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).
% max.strict_coboundedI1
tff(fact_4336_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).
% max.strict_coboundedI1
tff(fact_4337_max_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).
% max.strict_coboundedI1
tff(fact_4338_max_Ostrict__order__iff,axiom,
! [B: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(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_4339_max_Ostrict__order__iff,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(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_4340_max_Ostrict__order__iff,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
<=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_4341_max_Ostrict__order__iff,axiom,
! [B: num,A: num] :
( aa_num_o(aa_num_fun_num_o(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_4342_max_Ostrict__order__iff,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4343_max_Ostrict__order__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
<=> ( ( A = aa_int_int(ord_max_int(A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_4344_max_Ostrict__boundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
=> ~ ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
=> ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A) ) ) ).
% max.strict_boundedE
tff(fact_4345_max_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
=> ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).
% max.strict_boundedE
tff(fact_4346_max_Ostrict__boundedE,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),A)
=> ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).
% max.strict_boundedE
tff(fact_4347_max_Ostrict__boundedE,axiom,
! [B: num,C: num,A: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
=> ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).
% max.strict_boundedE
tff(fact_4348_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% max.strict_boundedE
tff(fact_4349_max_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(B),C)),A)
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% max.strict_boundedE
tff(fact_4350_less__max__iff__disj,axiom,
! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y))
<=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Xb)
| aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Y) ) ) ).
% less_max_iff_disj
tff(fact_4351_less__max__iff__disj,axiom,
! [Z: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y))
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
| aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).
% less_max_iff_disj
tff(fact_4352_less__max__iff__disj,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y))
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).
% less_max_iff_disj
tff(fact_4353_less__max__iff__disj,axiom,
! [Z: num,Xb: num,Y: num] :
( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
<=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xb)
| aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Y) ) ) ).
% less_max_iff_disj
tff(fact_4354_less__max__iff__disj,axiom,
! [Z: nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).
% less_max_iff_disj
tff(fact_4355_less__max__iff__disj,axiom,
! [Z: int,Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(ord_max_int(Xb),Y))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xb)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).
% less_max_iff_disj
tff(fact_4356_triangle__def,axiom,
! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% triangle_def
tff(fact_4357_even__succ__mod__exp,axiom,
! [A: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4358_even__succ__mod__exp,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( 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))),Nb)) = 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))),Nb))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4359_even__succ__mod__exp,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_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))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4360_divmod__algorithm__code_I6_J,axiom,
! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_de),unique5055182867167087721od_nat(Mb,Nb)) ).
% divmod_algorithm_code(6)
tff(fact_4361_divmod__algorithm__code_I6_J,axiom,
! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_df),unique5052692396658037445od_int(Mb,Nb)) ).
% divmod_algorithm_code(6)
tff(fact_4362_even__succ__div__exp,axiom,
! [A: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ) ) ) ).
% even_succ_div_exp
tff(fact_4363_even__succ__div__exp,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( 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))),Nb)) = 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))),Nb)) ) ) ) ).
% even_succ_div_exp
tff(fact_4364_even__succ__div__exp,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_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))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).
% even_succ_div_exp
tff(fact_4365_option_Osize__gen_I2_J,axiom,
! [Xb: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xb,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).
% option.size_gen(2)
tff(fact_4366_option_Osize__gen_I2_J,axiom,
! [Xb: fun_nat_nat,X2: nat] : size_option_nat(Xb,some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).
% option.size_gen(2)
tff(fact_4367_option_Osize__gen_I2_J,axiom,
! [Xb: fun_num_nat,X2: num] : size_option_num(Xb,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).
% option.size_gen(2)
tff(fact_4368_signed__take__bit__Suc,axiom,
! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_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(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% signed_take_bit_Suc
tff(fact_4369_nat__dvd__1__iff__1,axiom,
! [Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
<=> ( Mb = one_one_nat ) ) ).
% nat_dvd_1_iff_1
tff(fact_4370_dvd__1__iff__1,axiom,
! [Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
<=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).
% dvd_1_iff_1
tff(fact_4371_dvd__1__left,axiom,
! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).
% dvd_1_left
tff(fact_4372_dvd__add__triv__right__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).
% dvd_add_triv_right_iff
tff(fact_4373_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).
% dvd_add_triv_right_iff
tff(fact_4374_dvd__add__triv__right__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).
% dvd_add_triv_right_iff
tff(fact_4375_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).
% dvd_add_triv_right_iff
tff(fact_4376_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).
% dvd_add_triv_right_iff
tff(fact_4377_dvd__add__triv__left__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).
% dvd_add_triv_left_iff
tff(fact_4378_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).
% dvd_add_triv_left_iff
tff(fact_4379_dvd__add__triv__left__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).
% dvd_add_triv_left_iff
tff(fact_4380_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).
% dvd_add_triv_left_iff
tff(fact_4381_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).
% dvd_add_triv_left_iff
tff(fact_4382_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
<=> ( ( K = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).
% nat_mult_dvd_cancel_disj
tff(fact_4383_case__prod__conv,axiom,
! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) ).
% case_prod_conv
tff(fact_4384_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_4385_case__prod__conv,axiom,
! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).
% case_prod_conv
tff(fact_4386_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_4387_case__prod__conv,axiom,
! [F: fun_int_fun_int_o,A: int,B: int] :
( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).
% case_prod_conv
tff(fact_4388_dvd__mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( C = zero_z3403309356797280102nteger )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4389_dvd__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
<=> ( ( C = zero_zero_complex )
| aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4390_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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 )
| aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4391_dvd__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
<=> ( ( C = zero_zero_rat )
| aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4392_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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 )
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4393_dvd__mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( C = zero_z3403309356797280102nteger )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4394_dvd__mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
<=> ( ( C = zero_zero_complex )
| aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4395_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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 )
| aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4396_dvd__mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
<=> ( ( C = zero_zero_rat )
| aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4397_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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 )
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4398_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_4399_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_4400_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_4401_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_4402_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_4403_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_4404_unit__prod,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).
% unit_prod
tff(fact_4405_unit__prod,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).
% unit_prod
tff(fact_4406_unit__prod,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).
% unit_prod
tff(fact_4407_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4408_dvd__add__times__triv__left__iff,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(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))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4409_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),A)),B))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4410_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4411_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4412_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4413_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(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)))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4414_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),A)))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4415_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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)))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4416_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(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)))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4417_dvd__mult__div__cancel,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_4418_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4419_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_4420_dvd__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_4421_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4422_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_4423_unit__div__1__div__1,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4424_unit__div__1__div__1,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4425_unit__div__1__div__1,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_4426_unit__div__1__unit,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).
% unit_div_1_unit
tff(fact_4427_unit__div__1__unit,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(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_4428_unit__div__1__unit,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int) ) ).
% unit_div_1_unit
tff(fact_4429_unit__div,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).
% unit_div
tff(fact_4430_unit__div,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).
% unit_div
tff(fact_4431_unit__div,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).
% unit_div
tff(fact_4432_div__add,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).
% div_add
tff(fact_4433_div__add,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4434_div__add,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).
% div_add
tff(fact_4435_signed__take__bit__Suc__1,axiom,
! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).
% signed_take_bit_Suc_1
tff(fact_4436_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_4437_triangle__Suc,axiom,
! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).
% triangle_Suc
tff(fact_4438_even__Suc,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).
% even_Suc
tff(fact_4439_even__Suc__Suc__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).
% even_Suc_Suc_iff
tff(fact_4440_unit__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4441_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4442_unit__div__mult__self,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_4443_unit__mult__div__div,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4444_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4445_unit__mult__div__div,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).
% unit_mult_div_div
tff(fact_4446_pow__divides__pow__iff,axiom,
! [Nb: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).
% pow_divides_pow_iff
tff(fact_4447_pow__divides__pow__iff,axiom,
! [Nb: nat,A: int,B: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).
% pow_divides_pow_iff
tff(fact_4448_even__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4449_even__mult__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
| aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4450_even__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4451_odd__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
<=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).
% odd_add
tff(fact_4452_odd__add,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(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))
<=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).
% odd_add
tff(fact_4453_odd__add,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(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))
<=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).
% odd_add
tff(fact_4454_even__add,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).
% even_add
tff(fact_4455_even__add,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).
% even_add
tff(fact_4456_even__add,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).
% even_add
tff(fact_4457_even__mod__2__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).
% even_mod_2_iff
tff(fact_4458_even__mod__2__iff,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).
% even_mod_2_iff
tff(fact_4459_even__mod__2__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).
% even_mod_2_iff
tff(fact_4460_odd__Suc__div__two,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% odd_Suc_div_two
tff(fact_4461_even__Suc__div__two,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).
% even_Suc_div_two
tff(fact_4462_signed__take__bit__Suc__bit0,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_4463_dvd__numeral__simp,axiom,
! [Mb: num,Nb: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb))
<=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).
% dvd_numeral_simp
tff(fact_4464_dvd__numeral__simp,axiom,
! [Mb: num,Nb: num] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
<=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).
% dvd_numeral_simp
tff(fact_4465_dvd__numeral__simp,axiom,
! [Mb: num,Nb: num] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
<=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Mb)) ) ).
% dvd_numeral_simp
tff(fact_4466_zero__le__power__eq__numeral,axiom,
! [A: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_4467_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_4468_zero__le__power__eq__numeral,axiom,
! [A: real,W: num] :
( aa_real_o(aa_real_fun_real_o(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)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_4469_power__less__zero__eq,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).
% power_less_zero_eq
tff(fact_4470_power__less__zero__eq,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).
% power_less_zero_eq
tff(fact_4471_power__less__zero__eq,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).
% power_less_zero_eq
tff(fact_4472_power__less__zero__eq__numeral,axiom,
! [A: real,W: num] :
( aa_real_o(aa_real_fun_real_o(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)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).
% power_less_zero_eq_numeral
tff(fact_4473_power__less__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).
% power_less_zero_eq_numeral
tff(fact_4474_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).
% power_less_zero_eq_numeral
tff(fact_4475_even__plus__one__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4476_even__plus__one__iff,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4477_even__plus__one__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4478_even__diff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).
% even_diff
tff(fact_4479_even__diff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(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))
<=> aa_int_o(aa_int_fun_int_o(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_4480_odd__Suc__minus__one,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).
% odd_Suc_minus_one
tff(fact_4481_even__diff__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Mb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
| aa_nat_o(aa_nat_fun_nat_o(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,Mb),Nb)) ) ) ).
% even_diff_nat
tff(fact_4482_zero__less__power__eq__numeral,axiom,
! [A: real,W: num] :
( aa_real_o(aa_real_fun_real_o(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 )
| ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_4483_zero__less__power__eq__numeral,axiom,
! [A: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(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 )
| ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_4484_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( aa_int_o(aa_int_fun_int_o(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 )
| ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_4485_even__succ__div__2,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4486_even__succ__div__2,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4487_even__succ__div__2,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4488_even__succ__div__two,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4489_even__succ__div__two,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4490_even__succ__div__two,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4491_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4492_odd__succ__div__two,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(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_4493_odd__succ__div__two,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).
% odd_succ_div_two
tff(fact_4494_even__power,axiom,
! [A: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% even_power
tff(fact_4495_even__power,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% even_power
tff(fact_4496_even__power,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Nb))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).
% even_power
tff(fact_4497_odd__two__times__div__two__nat,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( 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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).
% odd_two_times_div_two_nat
tff(fact_4498_divmod__algorithm__code_I5_J,axiom,
! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dg),unique5055182867167087721od_nat(Mb,Nb)) ).
% divmod_algorithm_code(5)
tff(fact_4499_divmod__algorithm__code_I5_J,axiom,
! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_dh),unique5052692396658037445od_int(Mb,Nb)) ).
% divmod_algorithm_code(5)
tff(fact_4500_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4501_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(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_4502_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(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(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_4503_power__le__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
& ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
| ( aa_nat_o(aa_nat_fun_nat_o(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_4504_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
& ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
| ( aa_nat_o(aa_nat_fun_nat_o(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_4505_power__le__zero__eq__numeral,axiom,
! [A: real,W: num] :
( aa_real_o(aa_real_fun_real_o(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)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
& ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
| ( aa_nat_o(aa_nat_fun_nat_o(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_4506_semiring__parity__class_Oeven__mask__iff,axiom,
! [Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer))
<=> ( Nb = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4507_semiring__parity__class_Oeven__mask__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Nb)),one_one_nat))
<=> ( Nb = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4508_semiring__parity__class_Oeven__mask__iff,axiom,
! [Nb: nat] :
( aa_int_o(aa_int_fun_int_o(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))),Nb)),one_one_int))
<=> ( Nb = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4509_signed__take__bit__Suc__bit1,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_4510_prod_Ocase__distrib,axiom,
! [H: fun_nat_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_di(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4511_prod_Ocase__distrib,axiom,
! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
<=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_dj(H),F)),Prod) ) ).
% prod.case_distrib
tff(fact_4512_prod_Ocase__distrib,axiom,
! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
<=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_dk(H),F)),Prod) ) ).
% prod.case_distrib
tff(fact_4513_prod_Ocase__distrib,axiom,
! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_dl(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4514_prod_Ocase__distrib,axiom,
! [H: fun_na3970208099681795815_nat_o,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat7695548938202568488_nat_o(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun2100511394839469148_nat_o(aTP_Lamm_dm(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4515_prod_Ocase__distrib,axiom,
! [H: fun_fu1115759055147092071_o_nat,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun1432809138467554728_o_nat(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun3680403850923830236at_nat(aTP_Lamm_dn(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4516_prod_Ocase__distrib,axiom,
! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_do(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4517_prod_Ocase__distrib,axiom,
! [H: fun_na8713714802964096592at_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat7997280740862404369at_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aa_fun2484538639213223237at_nat(aTP_Lamm_dp(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4518_prod_Ocase__distrib,axiom,
! [H: fun_fu1413608398870722896at_nat,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] : aa_fun7582795104022253201at_nat(H,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun8817627472026143685at_nat(aTP_Lamm_dq(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4519_prod_Ocase__distrib,axiom,
! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_dr(H),F)),Prod) ).
% prod.case_distrib
tff(fact_4520_dvd__productE,axiom,
! [P2: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
=> ~ ! [X4: nat,Y3: nat] :
( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y3) )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X4),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y3),B) ) ) ) ).
% dvd_productE
tff(fact_4521_dvd__productE,axiom,
! [P2: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
=> ~ ! [X4: int,Y3: int] :
( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y3) )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X4),A)
=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y3),B) ) ) ) ).
% dvd_productE
tff(fact_4522_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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) )
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B7),B)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).
% division_decomp
tff(fact_4523_division__decomp,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(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) )
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B7),B)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).
% division_decomp
tff(fact_4524_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ~ ! [K3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).
% dvdE
tff(fact_4525_dvdE,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
=> ~ ! [K3: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).
% dvdE
tff(fact_4526_dvdE,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
=> ~ ! [K3: rat] : A != aa_rat_rat(times_times_rat(B),K3) ) ).
% dvdE
tff(fact_4527_dvdE,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ~ ! [K3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).
% dvdE
tff(fact_4528_dvdE,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ~ ! [K3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).
% dvdE
tff(fact_4529_dvdI,axiom,
! [A: code_integer,B: code_integer,K: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).
% dvdI
tff(fact_4530_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).
% dvdI
tff(fact_4531_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A = aa_rat_rat(times_times_rat(B),K) )
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).
% dvdI
tff(fact_4532_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).
% dvdI
tff(fact_4533_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).
% dvdI
tff(fact_4534_dvd__def,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
<=> ? [K2: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).
% dvd_def
tff(fact_4535_dvd__def,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
<=> ? [K2: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).
% dvd_def
tff(fact_4536_dvd__def,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
<=> ? [K2: rat] : A = aa_rat_rat(times_times_rat(B),K2) ) ).
% dvd_def
tff(fact_4537_dvd__def,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4538_dvd__def,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(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_4539_dvd__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).
% dvd_mult
tff(fact_4540_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).
% dvd_mult
tff(fact_4541_dvd__mult,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).
% dvd_mult
tff(fact_4542_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).
% dvd_mult
tff(fact_4543_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).
% dvd_mult
tff(fact_4544_dvd__mult2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).
% dvd_mult2
tff(fact_4545_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).
% dvd_mult2
tff(fact_4546_dvd__mult2,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).
% dvd_mult2
tff(fact_4547_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).
% dvd_mult2
tff(fact_4548_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).
% dvd_mult2
tff(fact_4549_dvd__mult__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).
% dvd_mult_left
tff(fact_4550_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).
% dvd_mult_left
tff(fact_4551_dvd__mult__left,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).
% dvd_mult_left
tff(fact_4552_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).
% dvd_mult_left
tff(fact_4553_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).
% dvd_mult_left
tff(fact_4554_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).
% dvd_triv_left
tff(fact_4555_dvd__triv__left,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).
% dvd_triv_left
tff(fact_4556_dvd__triv__left,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).
% dvd_triv_left
tff(fact_4557_dvd__triv__left,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).
% dvd_triv_left
tff(fact_4558_dvd__triv__left,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).
% dvd_triv_left
tff(fact_4559_mult__dvd__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4560_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
=> aa_real_o(aa_real_fun_real_o(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_4561_mult__dvd__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).
% mult_dvd_mono
tff(fact_4562_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
=> aa_nat_o(aa_nat_fun_nat_o(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_4563_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
=> aa_int_o(aa_int_fun_int_o(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_4564_dvd__mult__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).
% dvd_mult_right
tff(fact_4565_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).
% dvd_mult_right
tff(fact_4566_dvd__mult__right,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).
% dvd_mult_right
tff(fact_4567_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).
% dvd_mult_right
tff(fact_4568_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).
% dvd_mult_right
tff(fact_4569_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).
% dvd_triv_right
tff(fact_4570_dvd__triv__right,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).
% dvd_triv_right
tff(fact_4571_dvd__triv__right,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).
% dvd_triv_right
tff(fact_4572_dvd__triv__right,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).
% dvd_triv_right
tff(fact_4573_dvd__triv__right,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).
% dvd_triv_right
tff(fact_4574_dvd__unit__imp__unit,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).
% dvd_unit_imp_unit
tff(fact_4575_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).
% dvd_unit_imp_unit
tff(fact_4576_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).
% dvd_unit_imp_unit
tff(fact_4577_unit__imp__dvd,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).
% unit_imp_dvd
tff(fact_4578_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).
% unit_imp_dvd
tff(fact_4579_unit__imp__dvd,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).
% unit_imp_dvd
tff(fact_4580_one__dvd,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).
% one_dvd
tff(fact_4581_one__dvd,axiom,
! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),A) ).
% one_dvd
tff(fact_4582_one__dvd,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).
% one_dvd
tff(fact_4583_one__dvd,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).
% one_dvd
tff(fact_4584_one__dvd,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).
% one_dvd
tff(fact_4585_one__dvd,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).
% one_dvd
tff(fact_4586_dvd__add__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_add_right_iff
tff(fact_4587_dvd__add__right__iff,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).
% dvd_add_right_iff
tff(fact_4588_dvd__add__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).
% dvd_add_right_iff
tff(fact_4589_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_add_right_iff
tff(fact_4590_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_add_right_iff
tff(fact_4591_dvd__add__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).
% dvd_add_left_iff
tff(fact_4592_dvd__add__left__iff,axiom,
! [A: real,C: real,B: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).
% dvd_add_left_iff
tff(fact_4593_dvd__add__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).
% dvd_add_left_iff
tff(fact_4594_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).
% dvd_add_left_iff
tff(fact_4595_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).
% dvd_add_left_iff
tff(fact_4596_dvd__add,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).
% dvd_add
tff(fact_4597_dvd__add,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).
% dvd_add
tff(fact_4598_dvd__add,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)) ) ) ).
% dvd_add
tff(fact_4599_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).
% dvd_add
tff(fact_4600_dvd__add,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).
% dvd_add
tff(fact_4601_dvd__power__same,axiom,
! [Xb: code_integer,Y: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).
% dvd_power_same
tff(fact_4602_dvd__power__same,axiom,
! [Xb: nat,Y: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).
% dvd_power_same
tff(fact_4603_dvd__power__same,axiom,
! [Xb: real,Y: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).
% dvd_power_same
tff(fact_4604_dvd__power__same,axiom,
! [Xb: int,Y: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) ) ).
% dvd_power_same
tff(fact_4605_dvd__power__same,axiom,
! [Xb: complex,Y: complex,Nb: nat] :
( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ) ).
% dvd_power_same
tff(fact_4606_old_Oprod_Ocase,axiom,
! [F: fun_nat_fun_nat_nat,X1: nat,X2: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X1),X2) ).
% old.prod.case
tff(fact_4607_old_Oprod_Ocase,axiom,
! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).
% old.prod.case
tff(fact_4608_old_Oprod_Ocase,axiom,
! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).
% old.prod.case
tff(fact_4609_old_Oprod_Ocase,axiom,
! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).
% old.prod.case
tff(fact_4610_old_Oprod_Ocase,axiom,
! [F: fun_int_fun_int_o,X1: int,X2: int] :
( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
<=> aa_int_o(aa_int_fun_int_o(F,X1),X2) ) ).
% old.prod.case
tff(fact_4611_signed__take__bit__mult,axiom,
! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).
% signed_take_bit_mult
tff(fact_4612_dvd__diff__nat,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).
% dvd_diff_nat
tff(fact_4613_cond__case__prod__eta,axiom,
! [F: fun_nat_fun_nat_nat,G: fun_Pr2602117396471556196at_nat] :
( ! [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(F,X4),Y3) = aa_Pro4243173770044397973at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
=> ( produc6842872674320459806at_nat(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4614_cond__case__prod__eta,axiom,
! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
( ! [X4: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y3) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
=> ( produc4528391901023682385at_nat(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4615_cond__case__prod__eta,axiom,
! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
( ! [X4: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y3) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
=> ( produc6395135316500328296_nat_o(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4616_cond__case__prod__eta,axiom,
! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
( ! [X4: int,Y3: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y3) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3))
=> ( produc4245557441103728435nt_int(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4617_cond__case__prod__eta,axiom,
! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
( ! [X4: int,Y3: int] :
( aa_int_o(aa_int_fun_int_o(F,X4),Y3)
<=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3)) )
=> ( produc4947309494688390418_int_o(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4618_case__prod__eta,axiom,
! [F: fun_Pr2602117396471556196at_nat] : produc6842872674320459806at_nat(aTP_Lamm_ds(F)) = F ).
% case_prod_eta
tff(fact_4619_case__prod__eta,axiom,
! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_dt(F)) = F ).
% case_prod_eta
tff(fact_4620_case__prod__eta,axiom,
! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_du(F)) = F ).
% case_prod_eta
tff(fact_4621_case__prod__eta,axiom,
! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_dv(F)) = F ).
% case_prod_eta
tff(fact_4622_case__prod__eta,axiom,
! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_dw(F)) = F ).
% case_prod_eta
tff(fact_4623_case__prodE2,axiom,
! [Q: fun_nat_o,P: fun_nat_fun_nat_nat,Z: product_prod_nat_nat] :
( aa_nat_o(Q,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(P),Z))
=> ~ ! [X4: nat,Y3: nat] :
( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
=> ~ aa_nat_o(Q,aa_nat_nat(aa_nat_fun_nat_nat(P,X4),Y3)) ) ) ).
% case_prodE2
tff(fact_4624_case__prodE2,axiom,
! [Q: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z))
=> ~ ! [X4: nat,Y3: nat] :
( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
=> ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X4),Y3)) ) ) ).
% case_prodE2
tff(fact_4625_case__prodE2,axiom,
! [Q: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z))
=> ~ ! [X4: nat,Y3: nat] :
( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
=> ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X4),Y3)) ) ) ).
% case_prodE2
tff(fact_4626_case__prodE2,axiom,
! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
=> ~ ! [X4: int,Y3: int] :
( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
=> ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X4),Y3)) ) ) ).
% case_prodE2
tff(fact_4627_case__prodE2,axiom,
! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
=> ~ ! [X4: int,Y3: int] :
( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
=> ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X4),Y3)) ) ) ).
% case_prodE2
tff(fact_4628_strict__subset__divisors__dvd,axiom,
! [A: complex,B: complex] :
( aa_set_complex_o(ord_less_set_complex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dx,A))),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dx,B)))
<=> ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
& ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4629_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dy,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dy,B)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
& ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4630_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_dz,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_dz,B)))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4631_strict__subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ea,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ea,B)))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
& ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4632_even__signed__take__bit__iff,axiom,
! [Mb: nat,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).
% even_signed_take_bit_iff
tff(fact_4633_even__signed__take__bit__iff,axiom,
! [Mb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Mb),A))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).
% even_signed_take_bit_iff
tff(fact_4634_not__is__unit__0,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).
% not_is_unit_0
tff(fact_4635_not__is__unit__0,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).
% not_is_unit_0
tff(fact_4636_not__is__unit__0,axiom,
~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).
% not_is_unit_0
tff(fact_4637_minf_I10_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z2)
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2))
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2)) ) ) ).
% minf(10)
tff(fact_4638_minf_I10_J,axiom,
! [D: real,S2: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2))
<=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2)) ) ) ).
% minf(10)
tff(fact_4639_minf_I10_J,axiom,
! [D: rat,S2: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2))
<=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2)) ) ) ).
% minf(10)
tff(fact_4640_minf_I10_J,axiom,
! [D: nat,S2: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2)) ) ) ).
% minf(10)
tff(fact_4641_minf_I10_J,axiom,
! [D: int,S2: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2))
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2)) ) ) ).
% minf(10)
tff(fact_4642_minf_I9_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2)) ) ) ).
% minf(9)
tff(fact_4643_minf_I9_J,axiom,
! [D: real,S2: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2)) ) ) ).
% minf(9)
tff(fact_4644_minf_I9_J,axiom,
! [D: rat,S2: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
=> ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2)) ) ) ).
% minf(9)
tff(fact_4645_minf_I9_J,axiom,
! [D: nat,S2: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2)) ) ) ).
% minf(9)
tff(fact_4646_minf_I9_J,axiom,
! [D: int,S2: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2)) ) ) ).
% minf(9)
tff(fact_4647_pinf_I10_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X3)
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2))
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2)) ) ) ).
% pinf(10)
tff(fact_4648_pinf_I10_J,axiom,
! [D: real,S2: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2))
<=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2)) ) ) ).
% pinf(10)
tff(fact_4649_pinf_I10_J,axiom,
! [D: rat,S2: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2))
<=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2)) ) ) ).
% pinf(10)
tff(fact_4650_pinf_I10_J,axiom,
! [D: nat,S2: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2)) ) ) ).
% pinf(10)
tff(fact_4651_pinf_I10_J,axiom,
! [D: int,S2: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2))
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2)) ) ) ).
% pinf(10)
tff(fact_4652_pinf_I9_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S2)) ) ) ).
% pinf(9)
tff(fact_4653_pinf_I9_J,axiom,
! [D: real,S2: real] :
? [Z2: real] :
! [X3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S2)) ) ) ).
% pinf(9)
tff(fact_4654_pinf_I9_J,axiom,
! [D: rat,S2: rat] :
? [Z2: rat] :
! [X3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
=> ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S2)) ) ) ).
% pinf(9)
tff(fact_4655_pinf_I9_J,axiom,
! [D: nat,S2: nat] :
? [Z2: nat] :
! [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S2)) ) ) ).
% pinf(9)
tff(fact_4656_pinf_I9_J,axiom,
! [D: int,S2: int] :
? [Z2: int] :
! [X3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S2)) ) ) ).
% pinf(9)
tff(fact_4657_unit__mult__right__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4658_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4659_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(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_4660_unit__mult__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4661_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4662_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(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_4663_mult__unit__dvd__iff_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4664_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4665_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4666_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4667_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4668_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4669_mult__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% mult_unit_dvd_iff
tff(fact_4670_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% mult_unit_dvd_iff
tff(fact_4671_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% mult_unit_dvd_iff
tff(fact_4672_dvd__mult__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_mult_unit_iff
tff(fact_4673_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_mult_unit_iff
tff(fact_4674_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_mult_unit_iff
tff(fact_4675_is__unit__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).
% is_unit_mult_iff
tff(fact_4676_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).
% is_unit_mult_iff
tff(fact_4677_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).
% is_unit_mult_iff
tff(fact_4678_div__mult__div__if__dvd,axiom,
! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4679_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4680_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_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_4681_dvd__mult__imp__div,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).
% dvd_mult_imp_div
tff(fact_4682_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).
% dvd_mult_imp_div
tff(fact_4683_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)) ) ).
% dvd_mult_imp_div
tff(fact_4684_dvd__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4685_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4686_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4687_div__div__eq__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4688_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4689_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4690_div__mult__swap,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4691_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4692_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4693_dvd__div__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4694_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4695_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4696_dvd__div__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4697_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4698_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4699_div__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4700_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4701_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4702_unit__div__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4703_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4704_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4705_div__plus__div__distrib__dvd__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4706_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4707_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
=> ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).
% div_plus_div_distrib_dvd_left
tff(fact_4708_div__plus__div__distrib__dvd__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4709_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4710_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_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(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).
% div_plus_div_distrib_dvd_right
tff(fact_4711_div__power,axiom,
! [B: code_integer,A: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).
% div_power
tff(fact_4712_div__power,axiom,
! [B: nat,A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).
% div_power
tff(fact_4713_div__power,axiom,
! [B: int,A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_nat_int(power_power_int(aa_int_int(divide_divide_int(A),B)),Nb) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).
% div_power
tff(fact_4714_dvd__power__le,axiom,
! [Xb: code_integer,Y: code_integer,Nb: nat,Mb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Mb)) ) ) ).
% dvd_power_le
tff(fact_4715_dvd__power__le,axiom,
! [Xb: nat,Y: nat,Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).
% dvd_power_le
tff(fact_4716_dvd__power__le,axiom,
! [Xb: real,Y: real,Nb: nat,Mb: nat] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Mb)) ) ) ).
% dvd_power_le
tff(fact_4717_dvd__power__le,axiom,
! [Xb: int,Y: int,Nb: nat,Mb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Mb)) ) ) ).
% dvd_power_le
tff(fact_4718_dvd__power__le,axiom,
! [Xb: complex,Y: complex,Nb: nat,Mb: nat] :
( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Mb)) ) ) ).
% dvd_power_le
tff(fact_4719_power__le__dvd,axiom,
! [A: code_integer,Nb: nat,B: code_integer,Mb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),B) ) ) ).
% power_le_dvd
tff(fact_4720_power__le__dvd,axiom,
! [A: nat,Nb: nat,B: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).
% power_le_dvd
tff(fact_4721_power__le__dvd,axiom,
! [A: real,Nb: nat,B: real,Mb: nat] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B) ) ) ).
% power_le_dvd
tff(fact_4722_power__le__dvd,axiom,
! [A: int,Nb: nat,B: int,Mb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B) ) ) ).
% power_le_dvd
tff(fact_4723_power__le__dvd,axiom,
! [A: complex,Nb: nat,B: complex,Mb: nat] :
( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Nb)),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).
% power_le_dvd
tff(fact_4724_le__imp__power__dvd,axiom,
! [Mb: nat,Nb: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).
% le_imp_power_dvd
tff(fact_4725_le__imp__power__dvd,axiom,
! [Mb: nat,Nb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).
% le_imp_power_dvd
tff(fact_4726_le__imp__power__dvd,axiom,
! [Mb: nat,Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).
% le_imp_power_dvd
tff(fact_4727_le__imp__power__dvd,axiom,
! [Mb: nat,Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).
% le_imp_power_dvd
tff(fact_4728_le__imp__power__dvd,axiom,
! [Mb: nat,Nb: nat,A: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).
% le_imp_power_dvd
tff(fact_4729_dvd__pos__nat,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).
% dvd_pos_nat
tff(fact_4730_nat__dvd__not__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).
% nat_dvd_not_less
tff(fact_4731_dvd__minus__self,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
| aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).
% dvd_minus_self
tff(fact_4732_dvd__diffD,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).
% dvd_diffD
tff(fact_4733_dvd__diffD1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).
% dvd_diffD1
tff(fact_4734_less__eq__dvd__minus,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).
% less_eq_dvd_minus
tff(fact_4735_zdvd__not__zless,axiom,
! [Mb: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb)
=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb) ) ) ).
% zdvd_not_zless
tff(fact_4736_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),A)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),B)
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D4) ) ) ) ).
% bezout_add_nat
tff(fact_4737_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,Xb: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xb) = 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),Xb) = 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) ) )
=> ? [X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
& aa_nat_o(aa_nat_fun_nat_o(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),X4) = 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)),Y3)),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)),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).
% bezout_lemma_nat
tff(fact_4738_zdvd__mono,axiom,
! [K: int,Mb: int,Ta: int] :
( ( K != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Ta)) ) ) ).
% zdvd_mono
tff(fact_4739_zdvd__mult__cancel,axiom,
! [K: int,Mb: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb))
=> ( ( K != zero_zero_int )
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).
% zdvd_mult_cancel
tff(fact_4740_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),A)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),B)
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D4 )
| ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D4 ) ) ) ).
% bezout1_nat
tff(fact_4741_zdvd__period,axiom,
! [A: int,D: int,Xb: int,Ta: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).
% zdvd_period
tff(fact_4742_zdvd__reduce,axiom,
! [K: int,Nb: int,Mb: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).
% zdvd_reduce
tff(fact_4743_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).
% unit_dvdE
tff(fact_4744_unit__dvdE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ~ ( ( A != zero_zero_nat )
=> ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).
% unit_dvdE
tff(fact_4745_unit__dvdE,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ~ ( ( A != zero_zero_int )
=> ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).
% unit_dvdE
tff(fact_4746_unity__coeff__ex,axiom,
! [P: fun_Code_integer_o,L: code_integer] :
( ? [X: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X))
<=> ? [X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X,zero_z3403309356797280102nteger))
& aa_Code_integer_o(P,X) ) ) ).
% unity_coeff_ex
tff(fact_4747_unity__coeff__ex,axiom,
! [P: fun_complex_o,L: complex] :
( ? [X: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X))
<=> ? [X: complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X),zero_zero_complex))
& aa_complex_o(P,X) ) ) ).
% unity_coeff_ex
tff(fact_4748_unity__coeff__ex,axiom,
! [P: fun_real_o,L: real] :
( ? [X: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X))
<=> ? [X: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),zero_zero_real))
& aa_real_o(P,X) ) ) ).
% unity_coeff_ex
tff(fact_4749_unity__coeff__ex,axiom,
! [P: fun_rat_o,L: rat] :
( ? [X: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X))
<=> ? [X: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X),zero_zero_rat))
& aa_rat_o(P,X) ) ) ).
% unity_coeff_ex
tff(fact_4750_unity__coeff__ex,axiom,
! [P: fun_nat_o,L: nat] :
( ? [X: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X))
<=> ? [X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat))
& aa_nat_o(P,X) ) ) ).
% unity_coeff_ex
tff(fact_4751_unity__coeff__ex,axiom,
! [P: fun_int_o,L: int] :
( ? [X: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X))
<=> ? [X: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),zero_zero_int))
& aa_int_o(P,X) ) ) ).
% unity_coeff_ex
tff(fact_4752_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4753_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4754_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
=> ( ( aa_int_int(divide_divide_int(B),A) = aa_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_4755_dvd__div__iff__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4756_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4757_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4758_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4759_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4760_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4761_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4762_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4763_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( ( aa_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_4764_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4765_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4766_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4767_even__numeral,axiom,
! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,Nb))) ).
% even_numeral
tff(fact_4768_even__numeral,axiom,
! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).
% even_numeral
tff(fact_4769_even__numeral,axiom,
! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).
% even_numeral
tff(fact_4770_inf__period_I4_J,axiom,
! [D: code_integer,D5: code_integer,Ta: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D5)
=> ! [X3: code_integer,K4: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,Ta))
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D5)),Ta)) ) ) ).
% inf_period(4)
tff(fact_4771_inf__period_I4_J,axiom,
! [D: real,D5: real,Ta: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
=> ! [X3: real,K4: real] :
( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Ta))
<=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),Ta)) ) ) ).
% inf_period(4)
tff(fact_4772_inf__period_I4_J,axiom,
! [D: rat,D5: rat,Ta: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D5)
=> ! [X3: rat,K4: rat] :
( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),Ta))
<=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5))),Ta)) ) ) ).
% inf_period(4)
tff(fact_4773_inf__period_I4_J,axiom,
! [D: int,D5: int,Ta: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
=> ! [X3: int,K4: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
<=> ~ aa_int_o(aa_int_fun_int_o(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,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),Ta)) ) ) ).
% inf_period(4)
tff(fact_4774_inf__period_I3_J,axiom,
! [D: code_integer,D5: code_integer,Ta: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D5)
=> ! [X3: code_integer,K4: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,Ta))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D5)),Ta)) ) ) ).
% inf_period(3)
tff(fact_4775_inf__period_I3_J,axiom,
! [D: real,D5: real,Ta: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
=> ! [X3: real,K4: real] :
( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Ta))
<=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),Ta)) ) ) ).
% inf_period(3)
tff(fact_4776_inf__period_I3_J,axiom,
! [D: rat,D5: rat,Ta: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D5)
=> ! [X3: rat,K4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),Ta))
<=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5))),Ta)) ) ) ).
% inf_period(3)
tff(fact_4777_inf__period_I3_J,axiom,
! [D: int,D5: int,Ta: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
=> ! [X3: int,K4: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Ta))
<=> aa_int_o(aa_int_fun_int_o(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,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),Ta)) ) ) ).
% inf_period(3)
tff(fact_4778_is__unit__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4779_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4780_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4781_unit__div__mult__swap,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4782_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4783_unit__div__mult__swap,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_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_4784_unit__div__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4785_unit__div__commute,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4786_unit__div__commute,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4787_div__mult__unit2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4788_div__mult__unit2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4789_div__mult__unit2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4790_unit__eq__div2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4791_unit__eq__div2,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4792_unit__eq__div2,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( ( A = aa_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_4793_unit__eq__div1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4794_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4795_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( ( aa_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_4796_is__unit__power__iff,axiom,
! [A: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
| ( Nb = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4797_is__unit__power__iff,axiom,
! [A: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
| ( Nb = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4798_is__unit__power__iff,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
| ( Nb = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4799_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
tff(fact_4800_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4801_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).
% unit_imp_mod_eq_0
tff(fact_4802_dvd__imp__le,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).
% dvd_imp_le
tff(fact_4803_nat__mult__dvd__cancel1,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).
% nat_mult_dvd_cancel1
tff(fact_4804_dvd__mult__cancel,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).
% dvd_mult_cancel
tff(fact_4805_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X4: nat,Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),A)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),B)
& ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) ) ) ) ).
% bezout_add_strong_nat
tff(fact_4806_zdvd__imp__le,axiom,
! [Z: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).
% zdvd_imp_le
tff(fact_4807_mod__greater__zero__iff__not__dvd,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).
% mod_greater_zero_iff_not_dvd
tff(fact_4808_mod__eq__dvd__iff__nat,axiom,
! [Nb: nat,Mb: nat,Q3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).
% mod_eq_dvd_iff_nat
tff(fact_4809_finite__divisors__nat,axiom,
! [Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dy,Mb))) ) ).
% finite_divisors_nat
tff(fact_4810_even__zero,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).
% even_zero
tff(fact_4811_even__zero,axiom,
aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).
% even_zero
tff(fact_4812_even__zero,axiom,
aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).
% even_zero
tff(fact_4813_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B4: code_integer] :
( ( B4 != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4814_is__unitE,axiom,
! [A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ~ ( ( A != zero_zero_nat )
=> ! [B4: nat] :
( ( B4 != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4815_is__unitE,axiom,
! [A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ~ ( ( A != zero_zero_int )
=> ! [B4: int] :
( ( B4 != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B4),one_one_int)
=> ( ( aa_int_int(divide_divide_int(one_one_int),A) = B4 )
=> ( ( aa_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(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B4) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4816_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4817_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4818_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4819_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4820_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(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_4821_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4822_evenE,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4823_evenE,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4824_evenE,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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_4825_odd__one,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).
% odd_one
tff(fact_4826_odd__one,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).
% odd_one
tff(fact_4827_odd__one,axiom,
~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).
% odd_one
tff(fact_4828_odd__even__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).
% odd_even_add
tff(fact_4829_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
=> aa_nat_o(aa_nat_fun_nat_o(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_4830_odd__even__add,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
=> aa_int_o(aa_int_fun_int_o(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_4831_bit__eq__rec,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4832_bit__eq__rec,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
<=> aa_nat_o(aa_nat_fun_nat_o(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_4833_bit__eq__rec,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
& ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).
% bit_eq_rec
tff(fact_4834_dvd__power__iff,axiom,
! [Xb: code_integer,Mb: nat,Nb: nat] :
( ( Xb != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).
% dvd_power_iff
tff(fact_4835_dvd__power__iff,axiom,
! [Xb: nat,Mb: nat,Nb: nat] :
( ( Xb != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).
% dvd_power_iff
tff(fact_4836_dvd__power__iff,axiom,
! [Xb: int,Mb: nat,Nb: nat] :
( ( Xb != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),Nb))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).
% dvd_power_iff
tff(fact_4837_odd__numeral,axiom,
! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).
% odd_numeral
tff(fact_4838_odd__numeral,axiom,
! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).
% odd_numeral
tff(fact_4839_odd__numeral,axiom,
! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).
% odd_numeral
tff(fact_4840_dvd__power,axiom,
! [Nb: nat,Xb: code_integer] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
| ( Xb = one_one_Code_integer ) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) ) ).
% dvd_power
tff(fact_4841_dvd__power,axiom,
! [Nb: nat,Xb: rat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
| ( Xb = one_one_rat ) )
=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).
% dvd_power
tff(fact_4842_dvd__power,axiom,
! [Nb: nat,Xb: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
| ( Xb = one_one_nat ) )
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).
% dvd_power
tff(fact_4843_dvd__power,axiom,
! [Nb: nat,Xb: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
| ( Xb = one_one_real ) )
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).
% dvd_power
tff(fact_4844_dvd__power,axiom,
! [Nb: nat,Xb: int] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
| ( Xb = one_one_int ) )
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Nb)) ) ).
% dvd_power
tff(fact_4845_dvd__power,axiom,
! [Nb: nat,Xb: complex] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
| ( Xb = one_one_complex ) )
=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ).
% dvd_power
tff(fact_4846_div2__even__ext__nat,axiom,
! [Xb: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),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))) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
=> ( Xb = Y ) ) ) ).
% div2_even_ext_nat
tff(fact_4847_even__even__mod__4__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
<=> aa_nat_o(aa_nat_fun_nat_o(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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).
% even_even_mod_4_iff
tff(fact_4848_dvd__mult__cancel2,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
<=> ( Nb = one_one_nat ) ) ) ).
% dvd_mult_cancel2
tff(fact_4849_dvd__mult__cancel1,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
<=> ( Nb = one_one_nat ) ) ) ).
% dvd_mult_cancel1
tff(fact_4850_power__dvd__imp__le,axiom,
! [I: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% power_dvd_imp_le
tff(fact_4851_dvd__minus__add,axiom,
! [Q3: nat,Nb: nat,R2: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q3))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q3))) ) ) ) ).
% dvd_minus_add
tff(fact_4852_mod__nat__eqI,axiom,
! [R2: nat,Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).
% mod_nat_eqI
tff(fact_4853_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
| ( ( L = zero_zero_int )
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
| aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).
% mod_int_pos_iff
tff(fact_4854_even__two__times__div__two,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4855_even__two__times__div__two,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4856_even__two__times__div__two,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4857_even__iff__mod__2__eq__zero,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4858_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_4859_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
<=> ( aa_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_4860_odd__iff__mod__2__eq__one,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4861_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(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_4862_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
<=> ( aa_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_4863_power__mono__odd,axiom,
! [Nb: nat,A: rat,B: rat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).
% power_mono_odd
tff(fact_4864_power__mono__odd,axiom,
! [Nb: nat,A: int,B: int] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).
% power_mono_odd
tff(fact_4865_power__mono__odd,axiom,
! [Nb: nat,A: real,B: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).
% power_mono_odd
tff(fact_4866_odd__pos,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% odd_pos
tff(fact_4867_dvd__power__iff__le,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% dvd_power_iff_le
tff(fact_4868_even__unset__bit__iff,axiom,
! [Mb: nat,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
| ( Mb = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
tff(fact_4869_even__unset__bit__iff,axiom,
! [Mb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Mb),A))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
| ( Mb = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
tff(fact_4870_even__unset__bit__iff,axiom,
! [Mb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Mb),A))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
| ( Mb = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
tff(fact_4871_signed__take__bit__int__less__exp,axiom,
! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% signed_take_bit_int_less_exp
tff(fact_4872_even__set__bit__iff,axiom,
! [Mb: nat,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
& ( Mb != zero_zero_nat ) ) ) ).
% even_set_bit_iff
tff(fact_4873_even__set__bit__iff,axiom,
! [Mb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Mb),A))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
& ( Mb != zero_zero_nat ) ) ) ).
% even_set_bit_iff
tff(fact_4874_even__set__bit__iff,axiom,
! [Mb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Mb),A))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
& ( Mb != zero_zero_nat ) ) ) ).
% even_set_bit_iff
tff(fact_4875_even__flip__bit__iff,axiom,
! [Mb: nat,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
<=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
<=> ( Mb = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
tff(fact_4876_even__flip__bit__iff,axiom,
! [Mb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Mb),A))
<=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
<=> ( Mb = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
tff(fact_4877_even__flip__bit__iff,axiom,
! [Mb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Mb),A))
<=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
<=> ( Mb = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
tff(fact_4878_even__diff__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L))
<=> aa_int_o(aa_int_fun_int_o(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_4879_oddE,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4880_oddE,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(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_4881_oddE,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(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_4882_mod2__eq__if,axiom,
! [A: code_integer] :
modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).
% mod2_eq_if
tff(fact_4883_mod2__eq__if,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))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).
% mod2_eq_if
tff(fact_4884_mod2__eq__if,axiom,
! [A: int] :
aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).
% mod2_eq_if
tff(fact_4885_parity__cases,axiom,
! [A: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4886_parity__cases,axiom,
! [A: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(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_4887_parity__cases,axiom,
! [A: int] :
( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).
% parity_cases
tff(fact_4888_zero__le__power__eq,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).
% zero_le_power_eq
tff(fact_4889_zero__le__power__eq,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).
% zero_le_power_eq
tff(fact_4890_zero__le__power__eq,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).
% zero_le_power_eq
tff(fact_4891_zero__le__odd__power,axiom,
! [Nb: nat,A: rat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).
% zero_le_odd_power
tff(fact_4892_zero__le__odd__power,axiom,
! [Nb: nat,A: int] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).
% zero_le_odd_power
tff(fact_4893_zero__le__odd__power,axiom,
! [Nb: nat,A: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).
% zero_le_odd_power
tff(fact_4894_zero__le__even__power,axiom,
! [Nb: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).
% zero_le_even_power
tff(fact_4895_zero__le__even__power,axiom,
! [Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).
% zero_le_even_power
tff(fact_4896_zero__le__even__power,axiom,
! [Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).
% zero_le_even_power
tff(fact_4897_signed__take__bit__int__less__self__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),K)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).
% signed_take_bit_int_less_self_iff
tff(fact_4898_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).
% signed_take_bit_int_greater_eq_self_iff
tff(fact_4899_zero__less__power__eq,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
<=> ( ( Nb = zero_zero_nat )
| ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& ( A != zero_zero_real ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).
% zero_less_power_eq
tff(fact_4900_zero__less__power__eq,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
<=> ( ( Nb = zero_zero_nat )
| ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& ( A != zero_zero_rat ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).
% zero_less_power_eq
tff(fact_4901_zero__less__power__eq,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
<=> ( ( Nb = zero_zero_nat )
| ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& ( A != zero_zero_int ) )
| ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).
% zero_less_power_eq
tff(fact_4902_signed__take__bit__int__less__eq,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),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,Nb)))) ) ).
% signed_take_bit_int_less_eq
tff(fact_4903_even__mask__div__iff_H,axiom,
! [Mb: nat,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% even_mask_div_iff'
tff(fact_4904_even__mask__div__iff_H,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb)),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))),Nb)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% even_mask_div_iff'
tff(fact_4905_even__mask__div__iff_H,axiom,
! [Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% even_mask_div_iff'
tff(fact_4906_power__le__zero__eq,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
& ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
| ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_4907_power__le__zero__eq,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
& ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
| ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_4908_power__le__zero__eq,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
& ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
| ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_4909_option_Osize__gen_I1_J,axiom,
! [Xb: fun_nat_nat] : size_option_nat(Xb,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size_gen(1)
tff(fact_4910_option_Osize__gen_I1_J,axiom,
! [Xb: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xb,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size_gen(1)
tff(fact_4911_option_Osize__gen_I1_J,axiom,
! [Xb: fun_num_nat] : size_option_num(Xb,none_num) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size_gen(1)
tff(fact_4912_even__mod__4__div__2,axiom,
! [Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
=> aa_nat_o(aa_nat_fun_nat_o(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,Nb),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_4913_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_eb(L)),Qr) ).
% divmod_step_nat_def
tff(fact_4914_even__mask__div__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
<=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% even_mask_div_iff
tff(fact_4915_even__mask__div__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb)),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))),Nb)))
<=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% even_mask_div_iff
tff(fact_4916_even__mask__div__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
<=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).
% even_mask_div_iff
tff(fact_4917_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_ec(L)),Qr) ).
% divmod_step_int_def
tff(fact_4918_odd__mod__4__div__2,axiom,
! [Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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)) )
=> ~ aa_nat_o(aa_nat_fun_nat_o(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,Nb),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_4919_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,Mb: nat,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
| ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(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,Nb),Mb)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4920_even__mult__exp__div__exp__iff,axiom,
! [A: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
| ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& aa_nat_o(aa_nat_fun_nat_o(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,Nb),Mb)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4921_even__mult__exp__div__exp__iff,axiom,
! [A: int,Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
| ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_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,Nb),Mb)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4922_divmod__step__def,axiom,
! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_eb(L)),Qr) ).
% divmod_step_def
tff(fact_4923_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_ec(L)),Qr) ).
% divmod_step_def
tff(fact_4924_divmod__step__def,axiom,
! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ed(L),Qr) ).
% divmod_step_def
tff(fact_4925_infinite__growing,axiom,
! [X6: set_real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),X6)
=> ? [Xa: real] :
( aa_set_real_o(member_real(Xa),X6)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Xa) ) )
=> ~ aa_set_real_o(finite_finite_real,X6) ) ) ).
% infinite_growing
tff(fact_4926_infinite__growing,axiom,
! [X6: set_rat] :
( ( X6 != bot_bot_set_rat )
=> ( ! [X4: rat] :
( aa_set_rat_o(member_rat(X4),X6)
=> ? [Xa: rat] :
( aa_set_rat_o(member_rat(Xa),X6)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Xa) ) )
=> ~ aa_set_rat_o(finite_finite_rat,X6) ) ) ).
% infinite_growing
tff(fact_4927_infinite__growing,axiom,
! [X6: set_num] :
( ( X6 != bot_bot_set_num )
=> ( ! [X4: num] :
( aa_set_num_o(member_num(X4),X6)
=> ? [Xa: num] :
( aa_set_num_o(member_num(Xa),X6)
& aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Xa) ) )
=> ~ aa_set_num_o(finite_finite_num,X6) ) ) ).
% infinite_growing
tff(fact_4928_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),X6)
=> ? [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),X6)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa) ) )
=> ~ aa_set_nat_o(finite_finite_nat,X6) ) ) ).
% infinite_growing
tff(fact_4929_infinite__growing,axiom,
! [X6: set_int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),X6)
=> ? [Xa: int] :
( aa_set_int_o(member_int(Xa),X6)
& aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Xa) ) )
=> ~ aa_set_int_o(finite_finite_int,X6) ) ) ).
% infinite_growing
tff(fact_4930_ex__min__if__finite,axiom,
! [S3: set_real] :
( aa_set_real_o(finite_finite_real,S3)
=> ( ( S3 != bot_bot_set_real )
=> ? [X4: real] :
( aa_set_real_o(member_real(X4),S3)
& ~ ? [Xa: real] :
( aa_set_real_o(member_real(Xa),S3)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X4) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4931_ex__min__if__finite,axiom,
! [S3: set_rat] :
( aa_set_rat_o(finite_finite_rat,S3)
=> ( ( S3 != bot_bot_set_rat )
=> ? [X4: rat] :
( aa_set_rat_o(member_rat(X4),S3)
& ~ ? [Xa: rat] :
( aa_set_rat_o(member_rat(Xa),S3)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X4) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4932_ex__min__if__finite,axiom,
! [S3: set_num] :
( aa_set_num_o(finite_finite_num,S3)
=> ( ( S3 != bot_bot_set_num )
=> ? [X4: num] :
( aa_set_num_o(member_num(X4),S3)
& ~ ? [Xa: num] :
( aa_set_num_o(member_num(Xa),S3)
& aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X4) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4933_ex__min__if__finite,axiom,
! [S3: set_nat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( ( S3 != bot_bot_set_nat )
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
& ~ ? [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),S3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X4) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4934_ex__min__if__finite,axiom,
! [S3: set_int] :
( aa_set_int_o(finite_finite_int,S3)
=> ( ( S3 != bot_bot_set_int )
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),S3)
& ~ ? [Xa: int] :
( aa_set_int_o(member_int(Xa),S3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X4) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4935_vebt__buildup_Oelims,axiom,
! [Xb: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_buildup(Xb) = Y )
=> ( ( ( Xb = zero_zero_nat )
=> ( Y != vEBT_Leaf($false,$false) ) )
=> ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
=> ( Y != vEBT_Leaf($false,$false) ) )
=> ~ ! [Va2: nat] :
( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
=> ( Y != $ite(
aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
$let(
half: nat,
half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
$let(
half: nat,
half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
tff(fact_4936_divmod__nat__if,axiom,
! [Mb: nat,Nb: nat] :
divmod_nat(Mb,Nb) = $ite(
( ( Nb = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ee),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).
% divmod_nat_if
tff(fact_4937_signed__take__bit__Suc__minus__bit1,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_4938_signed__take__bit__rec,axiom,
! [Nb: nat,A: code_integer] :
bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),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,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).
% signed_take_bit_rec
tff(fact_4939_signed__take__bit__rec,axiom,
! [Nb: nat,A: int] :
aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_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(plus_plus_int,aa_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,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).
% signed_take_bit_rec
tff(fact_4940_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_4941_flip__bit__0,axiom,
! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4942_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_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4943_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_o(aa_nat_fun_nat_o(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_4944_set__decode__Suc,axiom,
! [Nb: nat,Xb: nat] :
( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xb))
<=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% set_decode_Suc
tff(fact_4945_intind,axiom,
! [I: nat,Nb: nat,P: fun_int_o,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
=> ( aa_int_o(P,Xb)
=> aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xb)),I)) ) ) ).
% intind
tff(fact_4946_intind,axiom,
! [I: nat,Nb: nat,P: fun_nat_o,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
=> ( aa_nat_o(P,Xb)
=> aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I)) ) ) ).
% intind
tff(fact_4947_intind,axiom,
! [I: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
=> ( aa_VEBT_VEBT_o(P,Xb)
=> aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I)) ) ) ).
% intind
tff(fact_4948_verit__minus__simplify_I4_J,axiom,
! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).
% verit_minus_simplify(4)
tff(fact_4949_verit__minus__simplify_I4_J,axiom,
! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).
% verit_minus_simplify(4)
tff(fact_4950_verit__minus__simplify_I4_J,axiom,
! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).
% verit_minus_simplify(4)
tff(fact_4951_verit__minus__simplify_I4_J,axiom,
! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).
% verit_minus_simplify(4)
tff(fact_4952_verit__minus__simplify_I4_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).
% verit_minus_simplify(4)
tff(fact_4953_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_4954_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_4955_add_Oinverse__inverse,axiom,
! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).
% add.inverse_inverse
tff(fact_4956_add_Oinverse__inverse,axiom,
! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).
% add.inverse_inverse
tff(fact_4957_add_Oinverse__inverse,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).
% add.inverse_inverse
tff(fact_4958_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_4959_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_4960_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_4961_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_4962_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_4963_case__prodI,axiom,
! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
( aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B))
=> aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B))) ) ).
% case_prodI
tff(fact_4964_case__prodI,axiom,
! [F: fun_num_fun_num_o,A: num,B: num] :
( aa_num_o(aa_num_fun_num_o(F,A),B)
=> aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B)) ) ).
% case_prodI
tff(fact_4965_case__prodI,axiom,
! [F: fun_nat_fun_num_o,A: nat,B: num] :
( aa_num_o(aa_nat_fun_num_o(F,A),B)
=> aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B)) ) ).
% case_prodI
tff(fact_4966_case__prodI,axiom,
! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) ) ).
% case_prodI
tff(fact_4967_case__prodI,axiom,
! [F: fun_int_fun_int_o,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(F,A),B)
=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).
% case_prodI
tff(fact_4968_case__prodI2,axiom,
! [P2: produc6271795597528267376eger_o,C: fun_Co480733329897759773un_o_o] :
( ! [A4: code_integer,B4: $o] :
( ( P2 = produc6677183202524767010eger_o(A4,(B4)) )
=> aa_o_o(aa_Cod6379977994291289310un_o_o(C,A4),(B4)) )
=> aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2) ) ).
% case_prodI2
tff(fact_4969_case__prodI2,axiom,
! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
( ! [A4: num,B4: num] :
( ( P2 = product_Pair_num_num(A4,B4) )
=> aa_num_o(aa_num_fun_num_o(C,A4),B4) )
=> aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).
% case_prodI2
tff(fact_4970_case__prodI2,axiom,
! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
( ! [A4: nat,B4: num] :
( ( P2 = product_Pair_nat_num(A4,B4) )
=> aa_num_o(aa_nat_fun_num_o(C,A4),B4) )
=> aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).
% case_prodI2
tff(fact_4971_case__prodI2,axiom,
! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
( ! [A4: nat,B4: nat] :
( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) )
=> aa_nat_o(aa_nat_fun_nat_o(C,A4),B4) )
=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).
% case_prodI2
tff(fact_4972_case__prodI2,axiom,
! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
( ! [A4: int,B4: int] :
( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
=> aa_int_o(aa_int_fun_int_o(C,A4),B4) )
=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).
% case_prodI2
tff(fact_4973_mem__case__prodI,axiom,
! [Z: complex,C: fun_Co5015990272074005067omplex,A: code_integer,B: $o] :
( aa_set_complex_o(member_complex(Z),aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A),(B)))
=> aa_set_complex_o(member_complex(Z),produc1043322548047392435omplex(C,produc6677183202524767010eger_o(A,(B)))) ) ).
% mem_case_prodI
tff(fact_4974_mem__case__prodI,axiom,
! [Z: real,C: fun_Co5368275265270938953t_real,A: code_integer,B: $o] :
( aa_set_real_o(member_real(Z),aa_o_set_real(aa_Cod7471715548787160626t_real(C,A),(B)))
=> aa_set_real_o(member_real(Z),produc242741666403216561t_real(C,produc6677183202524767010eger_o(A,(B)))) ) ).
% mem_case_prodI
tff(fact_4975_mem__case__prodI,axiom,
! [Z: nat,C: fun_Co7401227239307276909et_nat,A: code_integer,B: $o] :
( aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A),(B)))
=> aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,produc6677183202524767010eger_o(A,(B)))) ) ).
% mem_case_prodI
tff(fact_4976_mem__case__prodI,axiom,
! [Z: int,C: fun_Co2587678820519358153et_int,A: code_integer,B: $o] :
( aa_set_int_o(member_int(Z),aa_o_set_int(aa_Cod1361678317519647154et_int(C,A),(B)))
=> aa_set_int_o(member_int(Z),produc1253318751659547953et_int(C,produc6677183202524767010eger_o(A,(B)))) ) ).
% mem_case_prodI
tff(fact_4977_mem__case__prodI,axiom,
! [Z: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
( aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A),B))
=> aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,product_Pair_num_num(A,B))) ) ).
% mem_case_prodI
tff(fact_4978_mem__case__prodI,axiom,
! [Z: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
( aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A),B))
=> aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,product_Pair_num_num(A,B))) ) ).
% mem_case_prodI
tff(fact_4979_mem__case__prodI,axiom,
! [Z: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B))
=> aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(A,B))) ) ).
% mem_case_prodI
tff(fact_4980_mem__case__prodI,axiom,
! [Z: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
( aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A),B))
=> aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,product_Pair_num_num(A,B))) ) ).
% mem_case_prodI
tff(fact_4981_mem__case__prodI,axiom,
! [Z: complex,C: fun_na3925014923085203400omplex,A: nat,B: num] :
( aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A),B))
=> aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,product_Pair_nat_num(A,B))) ) ).
% mem_case_prodI
tff(fact_4982_mem__case__prodI,axiom,
! [Z: real,C: fun_na3666575854997362246t_real,A: nat,B: num] :
( aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B))
=> aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,product_Pair_nat_num(A,B))) ) ).
% mem_case_prodI
tff(fact_4983_mem__case__prodI2,axiom,
! [P2: produc6271795597528267376eger_o,Z: complex,C: fun_Co5015990272074005067omplex] :
( ! [A4: code_integer,B4: $o] :
( ( P2 = produc6677183202524767010eger_o(A4,(B4)) )
=> aa_set_complex_o(member_complex(Z),aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A4),(B4))) )
=> aa_set_complex_o(member_complex(Z),produc1043322548047392435omplex(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4984_mem__case__prodI2,axiom,
! [P2: produc6271795597528267376eger_o,Z: real,C: fun_Co5368275265270938953t_real] :
( ! [A4: code_integer,B4: $o] :
( ( P2 = produc6677183202524767010eger_o(A4,(B4)) )
=> aa_set_real_o(member_real(Z),aa_o_set_real(aa_Cod7471715548787160626t_real(C,A4),(B4))) )
=> aa_set_real_o(member_real(Z),produc242741666403216561t_real(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4985_mem__case__prodI2,axiom,
! [P2: produc6271795597528267376eger_o,Z: nat,C: fun_Co7401227239307276909et_nat] :
( ! [A4: code_integer,B4: $o] :
( ( P2 = produc6677183202524767010eger_o(A4,(B4)) )
=> aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A4),(B4))) )
=> aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4986_mem__case__prodI2,axiom,
! [P2: produc6271795597528267376eger_o,Z: int,C: fun_Co2587678820519358153et_int] :
( ! [A4: code_integer,B4: $o] :
( ( P2 = produc6677183202524767010eger_o(A4,(B4)) )
=> aa_set_int_o(member_int(Z),aa_o_set_int(aa_Cod1361678317519647154et_int(C,A4),(B4))) )
=> aa_set_int_o(member_int(Z),produc1253318751659547953et_int(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4987_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) )
=> aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A4),B4)) )
=> aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4988_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) )
=> aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A4),B4)) )
=> aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4989_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) )
=> aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A4),B4)) )
=> aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4990_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) )
=> aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A4),B4)) )
=> aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4991_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) )
=> aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A4),B4)) )
=> aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4992_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) )
=> aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A4),B4)) )
=> aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2)) ) ).
% mem_case_prodI2
tff(fact_4993_case__prodI2_H,axiom,
! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xb: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) = P2 )
=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A4),B4),Xb) )
=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Xb) ) ).
% case_prodI2'
tff(fact_4994_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% neg_le_iff_le
tff(fact_4995_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% neg_le_iff_le
tff(fact_4996_neg__le__iff__le,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% neg_le_iff_le
tff(fact_4997_neg__le__iff__le,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% neg_le_iff_le
tff(fact_4998_add_Oinverse__neutral,axiom,
aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
% add.inverse_neutral
tff(fact_4999_add_Oinverse__neutral,axiom,
aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).
% add.inverse_neutral
tff(fact_5000_add_Oinverse__neutral,axiom,
aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).
% add.inverse_neutral
tff(fact_5001_add_Oinverse__neutral,axiom,
aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).
% add.inverse_neutral
tff(fact_5002_add_Oinverse__neutral,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% add.inverse_neutral
tff(fact_5003_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_5004_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_5005_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_5006_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
<=> ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_5007_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_5008_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_5009_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_5010_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_5011_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
tff(fact_5012_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_5013_equal__neg__zero,axiom,
! [A: int] :
( ( A = aa_int_int(uminus_uminus_int,A) )
<=> ( A = zero_zero_int ) ) ).
% equal_neg_zero
tff(fact_5014_equal__neg__zero,axiom,
! [A: real] :
( ( A = aa_real_real(uminus_uminus_real,A) )
<=> ( A = zero_zero_real ) ) ).
% equal_neg_zero
tff(fact_5015_equal__neg__zero,axiom,
! [A: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,A) )
<=> ( A = zero_zero_rat ) ) ).
% equal_neg_zero
tff(fact_5016_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
tff(fact_5017_neg__equal__zero,axiom,
! [A: int] :
( ( aa_int_int(uminus_uminus_int,A) = A )
<=> ( A = zero_zero_int ) ) ).
% neg_equal_zero
tff(fact_5018_neg__equal__zero,axiom,
! [A: real] :
( ( aa_real_real(uminus_uminus_real,A) = A )
<=> ( A = zero_zero_real ) ) ).
% neg_equal_zero
tff(fact_5019_neg__equal__zero,axiom,
! [A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = A )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_zero
tff(fact_5020_neg__equal__zero,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
tff(fact_5021_neg__less__iff__less,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% neg_less_iff_less
tff(fact_5022_neg__less__iff__less,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).
% neg_less_iff_less
tff(fact_5023_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).
% neg_less_iff_less
tff(fact_5024_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% neg_less_iff_less
tff(fact_5025_neg__numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
<=> ( Mb = Nb ) ) ).
% neg_numeral_eq_iff
tff(fact_5026_neg__numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
<=> ( Mb = Nb ) ) ).
% neg_numeral_eq_iff
tff(fact_5027_neg__numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
<=> ( Mb = Nb ) ) ).
% neg_numeral_eq_iff
tff(fact_5028_neg__numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
<=> ( Mb = Nb ) ) ).
% neg_numeral_eq_iff
tff(fact_5029_neg__numeral__eq__iff,axiom,
! [Mb: num,Nb: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
<=> ( Mb = Nb ) ) ).
% neg_numeral_eq_iff
tff(fact_5030_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_5031_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_5032_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_5033_mult__minus__right,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).
% mult_minus_right
tff(fact_5034_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_5035_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_5036_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_5037_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_5038_minus__mult__minus,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).
% minus_mult_minus
tff(fact_5039_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_5040_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_5041_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_5042_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_5043_mult__minus__left,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).
% mult_minus_left
tff(fact_5044_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_5045_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_5046_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_5047_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_5048_add__minus__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).
% add_minus_cancel
tff(fact_5049_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_5050_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_5051_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_5052_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_5053_minus__add__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).
% minus_add_cancel
tff(fact_5054_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_5055_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_5056_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_5057_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_5058_minus__add__distrib,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).
% minus_add_distrib
tff(fact_5059_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_5060_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_5061_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_5062_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_5063_minus__diff__eq,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).
% minus_diff_eq
tff(fact_5064_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_5065_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
<=> ( (P)
=> (Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_5066_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
<=> ( (P)
=> (Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_5067_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q)))
<=> ( (P)
=> (Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_5068_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
<=> ( (P)
=> (Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_5069_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
<=> ( (P)
=> (Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_5070_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
<=> ( ~ (P)
& (Q) ) ) ).
% of_bool_less_iff
tff(fact_5071_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
<=> ( ~ (P)
& (Q) ) ) ).
% of_bool_less_iff
tff(fact_5072_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q)))
<=> ( ~ (P)
& (Q) ) ) ).
% of_bool_less_iff
tff(fact_5073_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
<=> ( ~ (P)
& (Q) ) ) ).
% of_bool_less_iff
tff(fact_5074_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
<=> ( ~ (P)
& (Q) ) ) ).
% of_bool_less_iff
tff(fact_5075_of__bool__eq_I2_J,axiom,
zero_n1201886186963655149omplex($true) = one_one_complex ).
% of_bool_eq(2)
tff(fact_5076_of__bool__eq_I2_J,axiom,
zero_n3304061248610475627l_real($true) = one_one_real ).
% of_bool_eq(2)
tff(fact_5077_of__bool__eq_I2_J,axiom,
zero_n2052037380579107095ol_rat($true) = one_one_rat ).
% of_bool_eq(2)
tff(fact_5078_of__bool__eq_I2_J,axiom,
zero_n2687167440665602831ol_nat($true) = one_one_nat ).
% of_bool_eq(2)
tff(fact_5079_of__bool__eq_I2_J,axiom,
aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).
% of_bool_eq(2)
tff(fact_5080_of__bool__eq_I2_J,axiom,
zero_n356916108424825756nteger($true) = one_one_Code_integer ).
% of_bool_eq(2)
tff(fact_5081_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_5082_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( zero_n3304061248610475627l_real((P)) = one_one_real )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_5083_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_5084_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( zero_n2687167440665602831ol_nat((P)) = one_one_nat )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_5085_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_5086_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_5087_replicate__eq__replicate,axiom,
! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Y) )
<=> ( ( Mb = Nb )
& ( ( Mb != zero_zero_nat )
=> ( Xb = Y ) ) ) ) ).
% replicate_eq_replicate
tff(fact_5088_length__replicate,axiom,
! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).
% length_replicate
tff(fact_5089_length__replicate,axiom,
! [Nb: nat,Xb: $o] : size_size_list_o(replicate_o(Nb,(Xb))) = Nb ).
% length_replicate
tff(fact_5090_length__replicate,axiom,
! [Nb: nat,Xb: nat] : size_size_list_nat(replicate_nat(Nb,Xb)) = Nb ).
% length_replicate
tff(fact_5091_length__replicate,axiom,
! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).
% length_replicate
tff(fact_5092_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_less_eq_nonneg
tff(fact_5093_neg__less__eq__nonneg,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% neg_less_eq_nonneg
tff(fact_5094_neg__less__eq__nonneg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% neg_less_eq_nonneg
tff(fact_5095_neg__less__eq__nonneg,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% neg_less_eq_nonneg
tff(fact_5096_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% less_eq_neg_nonpos
tff(fact_5097_less__eq__neg__nonpos,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% less_eq_neg_nonpos
tff(fact_5098_less__eq__neg__nonpos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% less_eq_neg_nonpos
tff(fact_5099_less__eq__neg__nonpos,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% less_eq_neg_nonpos
tff(fact_5100_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_le_0_iff_le
tff(fact_5101_neg__le__0__iff__le,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% neg_le_0_iff_le
tff(fact_5102_neg__le__0__iff__le,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% neg_le_0_iff_le
tff(fact_5103_neg__le__0__iff__le,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% neg_le_0_iff_le
tff(fact_5104_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% neg_0_le_iff_le
tff(fact_5105_neg__0__le__iff__le,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% neg_0_le_iff_le
tff(fact_5106_neg__0__le__iff__le,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% neg_0_le_iff_le
tff(fact_5107_neg__0__le__iff__le,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% neg_0_le_iff_le
tff(fact_5108_less__neg__neg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% less_neg_neg
tff(fact_5109_less__neg__neg,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% less_neg_neg
tff(fact_5110_less__neg__neg,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% less_neg_neg
tff(fact_5111_less__neg__neg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% less_neg_neg
tff(fact_5112_neg__less__pos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% neg_less_pos
tff(fact_5113_neg__less__pos,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).
% neg_less_pos
tff(fact_5114_neg__less__pos,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).
% neg_less_pos
tff(fact_5115_neg__less__pos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_less_pos
tff(fact_5116_neg__0__less__iff__less,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% neg_0_less_iff_less
tff(fact_5117_neg__0__less__iff__less,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% neg_0_less_iff_less
tff(fact_5118_neg__0__less__iff__less,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% neg_0_less_iff_less
tff(fact_5119_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% neg_0_less_iff_less
tff(fact_5120_neg__less__0__iff__less,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% neg_less_0_iff_less
tff(fact_5121_neg__less__0__iff__less,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).
% neg_less_0_iff_less
tff(fact_5122_neg__less__0__iff__less,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).
% neg_less_0_iff_less
tff(fact_5123_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_less_0_iff_less
tff(fact_5124_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_5125_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_5126_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_5127_ab__left__minus,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).
% ab_left_minus
tff(fact_5128_ab__left__minus,axiom,
! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).
% ab_left_minus
tff(fact_5129_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_5130_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_5131_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_5132_add_Oright__inverse,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).
% add.right_inverse
tff(fact_5133_add_Oright__inverse,axiom,
! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).
% add.right_inverse
tff(fact_5134_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_5135_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_5136_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_5137_verit__minus__simplify_I3_J,axiom,
! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).
% verit_minus_simplify(3)
tff(fact_5138_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% verit_minus_simplify(3)
tff(fact_5139_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_5140_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_5141_diff__0,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).
% diff_0
tff(fact_5142_diff__0,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).
% diff_0
tff(fact_5143_diff__0,axiom,
! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% diff_0
tff(fact_5144_add__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).
% add_neg_numeral_simps(3)
tff(fact_5145_add__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).
% add_neg_numeral_simps(3)
tff(fact_5146_add__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).
% add_neg_numeral_simps(3)
tff(fact_5147_add__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).
% add_neg_numeral_simps(3)
tff(fact_5148_add__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).
% add_neg_numeral_simps(3)
tff(fact_5149_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_5150_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_5151_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_5152_mult__minus1,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).
% mult_minus1
tff(fact_5153_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_5154_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_5155_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_5156_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_5157_mult__minus1__right,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).
% mult_minus1_right
tff(fact_5158_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_5159_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_5160_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_5161_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_5162_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).
% uminus_add_conv_diff
tff(fact_5163_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_5164_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_5165_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_5166_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_5167_diff__minus__eq__add,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).
% diff_minus_eq_add
tff(fact_5168_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_5169_divide__minus1,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).
% divide_minus1
tff(fact_5170_divide__minus1,axiom,
! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).
% divide_minus1
tff(fact_5171_divide__minus1,axiom,
! [Xb: rat] : aa_rat_rat(divide_divide_rat(Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xb) ).
% divide_minus1
tff(fact_5172_div__minus1__right,axiom,
! [A: int] : aa_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_5173_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_5174_zero__less__of__bool__iff,axiom,
! [P: $o] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
<=> (P) ) ).
% zero_less_of_bool_iff
tff(fact_5175_zero__less__of__bool__iff,axiom,
! [P: $o] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
<=> (P) ) ).
% zero_less_of_bool_iff
tff(fact_5176_zero__less__of__bool__iff,axiom,
! [P: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P)))
<=> (P) ) ).
% zero_less_of_bool_iff
tff(fact_5177_zero__less__of__bool__iff,axiom,
! [P: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
<=> (P) ) ).
% zero_less_of_bool_iff
tff(fact_5178_zero__less__of__bool__iff,axiom,
! [P: $o] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
<=> (P) ) ).
% zero_less_of_bool_iff
tff(fact_5179_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_5180_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_5181_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((P))),one_one_nat)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_5182_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_5183_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_5184_of__bool__not__iff,axiom,
! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex((P))) ).
% of_bool_not_iff
tff(fact_5185_of__bool__not__iff,axiom,
! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real((P))) ).
% of_bool_not_iff
tff(fact_5186_of__bool__not__iff,axiom,
! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).
% of_bool_not_iff
tff(fact_5187_of__bool__not__iff,axiom,
! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).
% of_bool_not_iff
tff(fact_5188_of__bool__not__iff,axiom,
! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).
% of_bool_not_iff
tff(fact_5189_Suc__0__mod__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = zero_n2687167440665602831ol_nat(Nb != aa_nat_nat(suc,zero_zero_nat)) ).
% Suc_0_mod_eq
tff(fact_5190_signed__take__bit__of__minus__1,axiom,
! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% signed_take_bit_of_minus_1
tff(fact_5191_signed__take__bit__of__minus__1,axiom,
! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),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_5192_in__set__replicate,axiom,
! [Xb: product_prod_nat_nat,Nb: nat,Y: product_prod_nat_nat] :
( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Nb,Y)))
<=> ( ( Xb = Y )
& ( Nb != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_5193_in__set__replicate,axiom,
! [Xb: complex,Nb: nat,Y: complex] :
( aa_set_complex_o(member_complex(Xb),set_complex2(replicate_complex(Nb,Y)))
<=> ( ( Xb = Y )
& ( Nb != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_5194_in__set__replicate,axiom,
! [Xb: real,Nb: nat,Y: real] :
( aa_set_real_o(member_real(Xb),set_real2(replicate_real(Nb,Y)))
<=> ( ( Xb = Y )
& ( Nb != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_5195_in__set__replicate,axiom,
! [Xb: int,Nb: nat,Y: int] :
( aa_set_int_o(member_int(Xb),set_int2(replicate_int(Nb,Y)))
<=> ( ( Xb = Y )
& ( Nb != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_5196_in__set__replicate,axiom,
! [Xb: nat,Nb: nat,Y: nat] :
( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Y)))
<=> ( ( Xb = Y )
& ( Nb != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_5197_in__set__replicate,axiom,
! [Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
<=> ( ( Xb = Y )
& ( Nb != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_5198_Bex__set__replicate,axiom,
! [Nb: nat,A: int,P: fun_int_o] :
( ? [X: int] :
( aa_set_int_o(member_int(X),set_int2(replicate_int(Nb,A)))
& aa_int_o(P,X) )
<=> ( aa_int_o(P,A)
& ( Nb != zero_zero_nat ) ) ) ).
% Bex_set_replicate
tff(fact_5199_Bex__set__replicate,axiom,
! [Nb: nat,A: nat,P: fun_nat_o] :
( ? [X: nat] :
( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Nb,A)))
& aa_nat_o(P,X) )
<=> ( aa_nat_o(P,A)
& ( Nb != zero_zero_nat ) ) ) ).
% Bex_set_replicate
tff(fact_5200_Bex__set__replicate,axiom,
! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
( ? [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
& aa_VEBT_VEBT_o(P,X) )
<=> ( aa_VEBT_VEBT_o(P,A)
& ( Nb != zero_zero_nat ) ) ) ).
% Bex_set_replicate
tff(fact_5201_Ball__set__replicate,axiom,
! [Nb: nat,A: int,P: fun_int_o] :
( ! [X: int] :
( aa_set_int_o(member_int(X),set_int2(replicate_int(Nb,A)))
=> aa_int_o(P,X) )
<=> ( aa_int_o(P,A)
| ( Nb = zero_zero_nat ) ) ) ).
% Ball_set_replicate
tff(fact_5202_Ball__set__replicate,axiom,
! [Nb: nat,A: nat,P: fun_nat_o] :
( ! [X: nat] :
( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Nb,A)))
=> aa_nat_o(P,X) )
<=> ( aa_nat_o(P,A)
| ( Nb = zero_zero_nat ) ) ) ).
% Ball_set_replicate
tff(fact_5203_Ball__set__replicate,axiom,
! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
=> aa_VEBT_VEBT_o(P,X) )
<=> ( aa_VEBT_VEBT_o(P,A)
| ( Nb = zero_zero_nat ) ) ) ).
% Ball_set_replicate
tff(fact_5204_pred__numeral__simps_I1_J,axiom,
aa_num_nat(pred_numeral,one) = zero_zero_nat ).
% pred_numeral_simps(1)
tff(fact_5205_Suc__eq__numeral,axiom,
! [Nb: nat,K: num] :
( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
<=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).
% Suc_eq_numeral
tff(fact_5206_eq__numeral__Suc,axiom,
! [K: num,Nb: nat] :
( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
<=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).
% eq_numeral_Suc
tff(fact_5207_nth__replicate,axiom,
! [I: nat,Nb: nat,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
=> ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),I) = Xb ) ) ).
% nth_replicate
tff(fact_5208_nth__replicate,axiom,
! [I: nat,Nb: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
=> ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I) = Xb ) ) ).
% nth_replicate
tff(fact_5209_nth__replicate,axiom,
! [I: nat,Nb: nat,Xb: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
=> ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I) = Xb ) ) ).
% nth_replicate
tff(fact_5210_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_5211_dbl__simps_I1_J,axiom,
! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).
% dbl_simps(1)
tff(fact_5212_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_5213_dbl__simps_I1_J,axiom,
! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).
% dbl_simps(1)
tff(fact_5214_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_5215_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_5216_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_5217_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_5218_dbl__inc__simps_I4_J,axiom,
aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% dbl_inc_simps(4)
tff(fact_5219_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_5220_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_5221_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_5222_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_5223_add__neg__numeral__special_I8_J,axiom,
aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).
% add_neg_numeral_special(8)
tff(fact_5224_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_5225_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_5226_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_5227_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_5228_add__neg__numeral__special_I7_J,axiom,
aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).
% add_neg_numeral_special(7)
tff(fact_5229_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_5230_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_5231_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_5232_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_5233_diff__numeral__special_I12_J,axiom,
aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).
% diff_numeral_special(12)
tff(fact_5234_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_5235_neg__one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
<=> ( Nb = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_5236_neg__one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
<=> ( Nb = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_5237_neg__one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
<=> ( Nb = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_5238_neg__one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
<=> ( Nb = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_5239_neg__one__eq__numeral__iff,axiom,
! [Nb: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
<=> ( Nb = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_5240_numeral__eq__neg__one__iff,axiom,
! [Nb: num] :
( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
<=> ( Nb = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_5241_numeral__eq__neg__one__iff,axiom,
! [Nb: num] :
( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
<=> ( Nb = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_5242_numeral__eq__neg__one__iff,axiom,
! [Nb: num] :
( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
<=> ( Nb = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_5243_numeral__eq__neg__one__iff,axiom,
! [Nb: num] :
( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
<=> ( Nb = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_5244_numeral__eq__neg__one__iff,axiom,
! [Nb: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
<=> ( Nb = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_5245_minus__one__mult__self,axiom,
! [Nb: 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)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).
% minus_one_mult_self
tff(fact_5246_minus__one__mult__self,axiom,
! [Nb: 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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).
% minus_one_mult_self
tff(fact_5247_minus__one__mult__self,axiom,
! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).
% minus_one_mult_self
tff(fact_5248_minus__one__mult__self,axiom,
! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).
% minus_one_mult_self
tff(fact_5249_minus__one__mult__self,axiom,
! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).
% minus_one_mult_self
tff(fact_5250_left__minus__one__mult__self,axiom,
! [Nb: 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)),Nb)),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)),Nb)),A)) = A ).
% left_minus_one_mult_self
tff(fact_5251_left__minus__one__mult__self,axiom,
! [Nb: 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)),Nb)),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)),Nb)),A)) = A ).
% left_minus_one_mult_self
tff(fact_5252_left__minus__one__mult__self,axiom,
! [Nb: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).
% left_minus_one_mult_self
tff(fact_5253_left__minus__one__mult__self,axiom,
! [Nb: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).
% left_minus_one_mult_self
tff(fact_5254_left__minus__one__mult__self,axiom,
! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).
% left_minus_one_mult_self
tff(fact_5255_mod__minus1__right,axiom,
! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% mod_minus1_right
tff(fact_5256_mod__minus1__right,axiom,
! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% mod_minus1_right
tff(fact_5257_max__number__of_I2_J,axiom,
! [U: num,V: num] :
ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).
% max_number_of(2)
tff(fact_5258_max__number__of_I2_J,axiom,
! [U: num,V: num] :
aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).
% max_number_of(2)
tff(fact_5259_max__number__of_I2_J,axiom,
! [U: num,V: num] :
aa_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))) = $ite(aa_int_o(aa_int_fun_int_o(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(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).
% max_number_of(2)
tff(fact_5260_max__number__of_I2_J,axiom,
! [U: num,V: num] :
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,U)) ).
% max_number_of(2)
tff(fact_5261_max__number__of_I3_J,axiom,
! [U: num,V: num] :
ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).
% max_number_of(3)
tff(fact_5262_max__number__of_I3_J,axiom,
! [U: num,V: num] :
aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).
% max_number_of(3)
tff(fact_5263_max__number__of_I3_J,axiom,
! [U: num,V: num] :
aa_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)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),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_5264_max__number__of_I3_J,axiom,
! [U: num,V: num] :
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).
% max_number_of(3)
tff(fact_5265_max__number__of_I4_J,axiom,
! [U: num,V: num] :
ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).
% max_number_of(4)
tff(fact_5266_max__number__of_I4_J,axiom,
! [U: num,V: num] :
aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).
% max_number_of(4)
tff(fact_5267_max__number__of_I4_J,axiom,
! [U: num,V: num] :
aa_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))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).
% max_number_of(4)
tff(fact_5268_max__number__of_I4_J,axiom,
! [U: num,V: num] :
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).
% max_number_of(4)
tff(fact_5269_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_5270_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,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).
% semiring_norm(168)
tff(fact_5271_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_5272_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).
% semiring_norm(168)
tff(fact_5273_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_5274_diff__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% diff_numeral_simps(3)
tff(fact_5275_diff__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% diff_numeral_simps(3)
tff(fact_5276_diff__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% diff_numeral_simps(3)
tff(fact_5277_diff__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% diff_numeral_simps(3)
tff(fact_5278_diff__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).
% diff_numeral_simps(3)
tff(fact_5279_diff__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% diff_numeral_simps(2)
tff(fact_5280_diff__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% diff_numeral_simps(2)
tff(fact_5281_diff__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% diff_numeral_simps(2)
tff(fact_5282_diff__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% diff_numeral_simps(2)
tff(fact_5283_diff__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).
% diff_numeral_simps(2)
tff(fact_5284_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_5285_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,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_5286_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_5287_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_5288_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_5289_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_5290_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_5291_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_5292_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_5293_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_5294_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_5295_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,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_5296_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_5297_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_5298_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_5299_mult__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(3)
tff(fact_5300_mult__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(3)
tff(fact_5301_mult__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(3)
tff(fact_5302_mult__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(3)
tff(fact_5303_mult__neg__numeral__simps_I3_J,axiom,
! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(3)
tff(fact_5304_mult__neg__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(2)
tff(fact_5305_mult__neg__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(2)
tff(fact_5306_mult__neg__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(2)
tff(fact_5307_mult__neg__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(2)
tff(fact_5308_mult__neg__numeral__simps_I2_J,axiom,
! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).
% mult_neg_numeral_simps(2)
tff(fact_5309_mult__neg__numeral__simps_I1_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% mult_neg_numeral_simps(1)
tff(fact_5310_mult__neg__numeral__simps_I1_J,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% mult_neg_numeral_simps(1)
tff(fact_5311_mult__neg__numeral__simps_I1_J,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% mult_neg_numeral_simps(1)
tff(fact_5312_mult__neg__numeral__simps_I1_J,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% mult_neg_numeral_simps(1)
tff(fact_5313_mult__neg__numeral__simps_I1_J,axiom,
! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).
% mult_neg_numeral_simps(1)
tff(fact_5314_less__Suc__numeral,axiom,
! [Nb: nat,K: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).
% less_Suc_numeral
tff(fact_5315_less__numeral__Suc,axiom,
! [K: num,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).
% less_numeral_Suc
tff(fact_5316_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_5317_le__numeral__Suc,axiom,
! [K: num,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).
% le_numeral_Suc
tff(fact_5318_le__Suc__numeral,axiom,
! [Nb: nat,K: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).
% le_Suc_numeral
tff(fact_5319_neg__numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).
% neg_numeral_le_iff
tff(fact_5320_neg__numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).
% neg_numeral_le_iff
tff(fact_5321_neg__numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).
% neg_numeral_le_iff
tff(fact_5322_neg__numeral__le__iff,axiom,
! [Mb: num,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).
% neg_numeral_le_iff
tff(fact_5323_diff__Suc__numeral,axiom,
! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).
% diff_Suc_numeral
tff(fact_5324_diff__numeral__Suc,axiom,
! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).
% diff_numeral_Suc
tff(fact_5325_neg__numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).
% neg_numeral_less_iff
tff(fact_5326_neg__numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).
% neg_numeral_less_iff
tff(fact_5327_neg__numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).
% neg_numeral_less_iff
tff(fact_5328_neg__numeral__less__iff,axiom,
! [Mb: num,Nb: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).
% neg_numeral_less_iff
tff(fact_5329_max__Suc__numeral,axiom,
! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).
% max_Suc_numeral
tff(fact_5330_max__numeral__Suc,axiom,
! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).
% max_numeral_Suc
tff(fact_5331_not__neg__one__le__neg__numeral__iff,axiom,
! [Mb: num] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)))
<=> ( Mb != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_5332_not__neg__one__le__neg__numeral__iff,axiom,
! [Mb: num] :
( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)))
<=> ( Mb != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_5333_not__neg__one__le__neg__numeral__iff,axiom,
! [Mb: num] :
( ~ aa_int_o(aa_int_fun_int_o(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,Mb)))
<=> ( Mb != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_5334_not__neg__one__le__neg__numeral__iff,axiom,
! [Mb: num] :
( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)))
<=> ( Mb != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_5335_neg__numeral__less__neg__one__iff,axiom,
! [Mb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
<=> ( Mb != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_5336_neg__numeral__less__neg__one__iff,axiom,
! [Mb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real))
<=> ( Mb != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_5337_neg__numeral__less__neg__one__iff,axiom,
! [Mb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
<=> ( Mb != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_5338_neg__numeral__less__neg__one__iff,axiom,
! [Mb: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
<=> ( Mb != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_5339_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).
% le_divide_eq_numeral1(2)
tff(fact_5340_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).
% le_divide_eq_numeral1(2)
tff(fact_5341_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).
% divide_le_eq_numeral1(2)
tff(fact_5342_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).
% divide_le_eq_numeral1(2)
tff(fact_5343_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,aa_num_real(numeral_numeral_real,W))) = A )
<=> $ite(aa_real_real(uminus_uminus_real,aa_num_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,aa_num_real(numeral_numeral_real,W))),A = zero_zero_real) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_5344_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 )
<=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_5345_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
<=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_5346_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,aa_num_real(numeral_numeral_real,W))) )
<=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_5347_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))) )
<=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_5348_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
<=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_5349_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).
% less_divide_eq_numeral1(2)
tff(fact_5350_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).
% less_divide_eq_numeral1(2)
tff(fact_5351_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).
% divide_less_eq_numeral1(2)
tff(fact_5352_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).
% divide_less_eq_numeral1(2)
tff(fact_5353_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_5354_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_5355_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_5356_power2__minus,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_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_5357_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_5358_odd__of__bool__self,axiom,
! [P2: $o] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_n2687167440665602831ol_nat((P2)))
<=> (P2) ) ).
% odd_of_bool_self
tff(fact_5359_odd__of__bool__self,axiom,
! [P2: $o] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
<=> (P2) ) ).
% odd_of_bool_self
tff(fact_5360_odd__of__bool__self,axiom,
! [P2: $o] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
<=> (P2) ) ).
% odd_of_bool_self
tff(fact_5361_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_5362_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_5363_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_5364_add__neg__numeral__special_I9_J,axiom,
aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_5365_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_5366_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_5367_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5368_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_5369_diff__numeral__special_I10_J,axiom,
aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5370_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_5371_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_5372_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)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5373_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_5374_diff__numeral__special_I11_J,axiom,
aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5375_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_5376_minus__1__div__2__eq,axiom,
aa_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_5377_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_5378_minus__1__mod__2__eq,axiom,
aa_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_5379_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_5380_bits__minus__1__mod__2__eq,axiom,
aa_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_5381_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_5382_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,Nb: 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))),Nb)) = 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))),Nb)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5383_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,Nb: 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))),Nb)) = 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))),Nb)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5384_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: complex,Nb: 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))),Nb)) = 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))),Nb)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5385_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_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))),Nb)) = 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))),Nb)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5386_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,Nb: 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))),Nb)) = 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))),Nb)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5387_of__bool__half__eq__0,axiom,
! [B: $o] : 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_5388_of__bool__half__eq__0,axiom,
! [B: $o] : aa_int_int(divide_divide_int(aa_o_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_5389_of__bool__half__eq__0,axiom,
! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).
% of_bool_half_eq_0
tff(fact_5390_Parity_Oring__1__class_Opower__minus__even,axiom,
! [Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5391_Parity_Oring__1__class_Opower__minus__even,axiom,
! [Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5392_Parity_Oring__1__class_Opower__minus__even,axiom,
! [Nb: nat,A: complex] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5393_Parity_Oring__1__class_Opower__minus__even,axiom,
! [Nb: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5394_Parity_Oring__1__class_Opower__minus__even,axiom,
! [Nb: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5395_power__minus__odd,axiom,
! [Nb: nat,A: int] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb)) ) ) ).
% power_minus_odd
tff(fact_5396_power__minus__odd,axiom,
! [Nb: nat,A: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).
% power_minus_odd
tff(fact_5397_power__minus__odd,axiom,
! [Nb: nat,A: complex] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb)) ) ) ).
% power_minus_odd
tff(fact_5398_power__minus__odd,axiom,
! [Nb: nat,A: rat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb)) ) ) ).
% power_minus_odd
tff(fact_5399_power__minus__odd,axiom,
! [Nb: nat,A: code_integer] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).
% power_minus_odd
tff(fact_5400_diff__numeral__special_I3_J,axiom,
! [Nb: 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,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% diff_numeral_special(3)
tff(fact_5401_diff__numeral__special_I3_J,axiom,
! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% diff_numeral_special(3)
tff(fact_5402_diff__numeral__special_I3_J,axiom,
! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% diff_numeral_special(3)
tff(fact_5403_diff__numeral__special_I3_J,axiom,
! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% diff_numeral_special(3)
tff(fact_5404_diff__numeral__special_I3_J,axiom,
! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).
% diff_numeral_special(3)
tff(fact_5405_diff__numeral__special_I4_J,axiom,
! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),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,Mb),one))) ).
% diff_numeral_special(4)
tff(fact_5406_diff__numeral__special_I4_J,axiom,
! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).
% diff_numeral_special(4)
tff(fact_5407_diff__numeral__special_I4_J,axiom,
! [Mb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).
% diff_numeral_special(4)
tff(fact_5408_diff__numeral__special_I4_J,axiom,
! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).
% diff_numeral_special(4)
tff(fact_5409_diff__numeral__special_I4_J,axiom,
! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).
% diff_numeral_special(4)
tff(fact_5410_set__decode__0,axiom,
! [Xb: nat] :
( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xb))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).
% set_decode_0
tff(fact_5411_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_5412_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5413_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_5414_dbl__simps_I4_J,axiom,
aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5415_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_5416_power__minus1__even,axiom,
! [Nb: 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))),Nb)) = one_one_int ).
% power_minus1_even
tff(fact_5417_power__minus1__even,axiom,
! [Nb: 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))),Nb)) = one_one_real ).
% power_minus1_even
tff(fact_5418_power__minus1__even,axiom,
! [Nb: 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))),Nb)) = one_one_complex ).
% power_minus1_even
tff(fact_5419_power__minus1__even,axiom,
! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_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))),Nb)) = one_one_rat ).
% power_minus1_even
tff(fact_5420_power__minus1__even,axiom,
! [Nb: 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))),Nb)) = one_one_Code_integer ).
% power_minus1_even
tff(fact_5421_neg__one__even__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).
% neg_one_even_power
tff(fact_5422_neg__one__even__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).
% neg_one_even_power
tff(fact_5423_neg__one__even__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).
% neg_one_even_power
tff(fact_5424_neg__one__even__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).
% neg_one_even_power
tff(fact_5425_neg__one__even__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).
% neg_one_even_power
tff(fact_5426_neg__one__odd__power,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).
% neg_one_odd_power
tff(fact_5427_neg__one__odd__power,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).
% neg_one_odd_power
tff(fact_5428_neg__one__odd__power,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).
% neg_one_odd_power
tff(fact_5429_neg__one__odd__power,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).
% neg_one_odd_power
tff(fact_5430_neg__one__odd__power,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).
% neg_one_odd_power
tff(fact_5431_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_5432_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(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% signed_take_bit_0
tff(fact_5433_signed__take__bit__Suc__minus__bit0,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_5434_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_5435_one__div__2__pow__eq,axiom,
! [Nb: 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))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_5436_one__div__2__pow__eq,axiom,
! [Nb: nat] : aa_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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_5437_one__div__2__pow__eq,axiom,
! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_5438_bits__1__div__exp,axiom,
! [Nb: 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))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_5439_bits__1__div__exp,axiom,
! [Nb: nat] : aa_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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_5440_bits__1__div__exp,axiom,
! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_5441_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_5442_one__mod__2__pow__eq,axiom,
! [Nb: 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))),Nb)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% one_mod_2_pow_eq
tff(fact_5443_one__mod__2__pow__eq,axiom,
! [Nb: nat] : aa_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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% one_mod_2_pow_eq
tff(fact_5444_one__mod__2__pow__eq,axiom,
! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% one_mod_2_pow_eq
tff(fact_5445_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_5446_dvd__antisym,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
=> ( Mb = Nb ) ) ) ).
% dvd_antisym
tff(fact_5447_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).
% verit_negate_coefficient(3)
tff(fact_5448_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).
% verit_negate_coefficient(3)
tff(fact_5449_verit__negate__coefficient_I3_J,axiom,
! [A: rat,B: rat] :
( ( A = B )
=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).
% verit_negate_coefficient(3)
tff(fact_5450_verit__negate__coefficient_I3_J,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).
% verit_negate_coefficient(3)
tff(fact_5451_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_5452_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_5453_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_5454_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% equation_minus_iff
tff(fact_5455_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_5456_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_5457_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_5458_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_5459_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = B )
<=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).
% minus_equation_iff
tff(fact_5460_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_5461_of__bool__conj,axiom,
! [P: $o,Q: $o] :
zero_n3304061248610475627l_real(( (P)
& (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).
% of_bool_conj
tff(fact_5462_of__bool__conj,axiom,
! [P: $o,Q: $o] :
zero_n2052037380579107095ol_rat(( (P)
& (Q) )) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).
% of_bool_conj
tff(fact_5463_of__bool__conj,axiom,
! [P: $o,Q: $o] :
zero_n2687167440665602831ol_nat(( (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_5464_of__bool__conj,axiom,
! [P: $o,Q: $o] :
aa_o_int(zero_n2684676970156552555ol_int,
( (P)
& (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).
% of_bool_conj
tff(fact_5465_of__bool__conj,axiom,
! [P: $o,Q: $o] :
zero_n356916108424825756nteger(( (P)
& (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).
% of_bool_conj
tff(fact_5466_mem__case__prodE,axiom,
! [Z: complex,C: fun_Co5015990272074005067omplex,P2: produc6271795597528267376eger_o] :
( aa_set_complex_o(member_complex(Z),produc1043322548047392435omplex(C,P2))
=> ~ ! [X4: code_integer,Y3: $o] :
( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
=> ~ aa_set_complex_o(member_complex(Z),aa_o_set_complex(aa_Cod2779864889637189172omplex(C,X4),(Y3))) ) ) ).
% mem_case_prodE
tff(fact_5467_mem__case__prodE,axiom,
! [Z: real,C: fun_Co5368275265270938953t_real,P2: produc6271795597528267376eger_o] :
( aa_set_real_o(member_real(Z),produc242741666403216561t_real(C,P2))
=> ~ ! [X4: code_integer,Y3: $o] :
( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
=> ~ aa_set_real_o(member_real(Z),aa_o_set_real(aa_Cod7471715548787160626t_real(C,X4),(Y3))) ) ) ).
% mem_case_prodE
tff(fact_5468_mem__case__prodE,axiom,
! [Z: nat,C: fun_Co7401227239307276909et_nat,P2: produc6271795597528267376eger_o] :
( aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2))
=> ~ ! [X4: code_integer,Y3: $o] :
( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
=> ~ aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,X4),(Y3))) ) ) ).
% mem_case_prodE
tff(fact_5469_mem__case__prodE,axiom,
! [Z: int,C: fun_Co2587678820519358153et_int,P2: produc6271795597528267376eger_o] :
( aa_set_int_o(member_int(Z),produc1253318751659547953et_int(C,P2))
=> ~ ! [X4: code_integer,Y3: $o] :
( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
=> ~ aa_set_int_o(member_int(Z),aa_o_set_int(aa_Cod1361678317519647154et_int(C,X4),(Y3))) ) ) ).
% mem_case_prodE
tff(fact_5470_mem__case__prodE,axiom,
! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
( aa_set_complex_o(member_complex(Z),produc2866383454006189126omplex(C,P2))
=> ~ ! [X4: num,Y3: num] :
( ( P2 = product_Pair_num_num(X4,Y3) )
=> ~ aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,X4),Y3)) ) ) ).
% mem_case_prodE
tff(fact_5471_mem__case__prodE,axiom,
! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
( aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2))
=> ~ ! [X4: num,Y3: num] :
( ( P2 = product_Pair_num_num(X4,Y3) )
=> ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,X4),Y3)) ) ) ).
% mem_case_prodE
tff(fact_5472_mem__case__prodE,axiom,
! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
=> ~ ! [X4: num,Y3: num] :
( ( P2 = product_Pair_num_num(X4,Y3) )
=> ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X4),Y3)) ) ) ).
% mem_case_prodE
tff(fact_5473_mem__case__prodE,axiom,
! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
( aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2))
=> ~ ! [X4: num,Y3: num] :
( ( P2 = product_Pair_num_num(X4,Y3) )
=> ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,X4),Y3)) ) ) ).
% mem_case_prodE
tff(fact_5474_mem__case__prodE,axiom,
! [Z: complex,C: fun_na3925014923085203400omplex,P2: product_prod_nat_num] :
( aa_set_complex_o(member_complex(Z),produc6231982587499038204omplex(C,P2))
=> ~ ! [X4: nat,Y3: num] :
( ( P2 = product_Pair_nat_num(X4,Y3) )
=> ~ aa_set_complex_o(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,X4),Y3)) ) ) ).
% mem_case_prodE
tff(fact_5475_mem__case__prodE,axiom,
! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
( aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2))
=> ~ ! [X4: nat,Y3: num] :
( ( P2 = product_Pair_nat_num(X4,Y3) )
=> ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,X4),Y3)) ) ) ).
% mem_case_prodE
tff(fact_5476_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% le_minus_iff
tff(fact_5477_le__minus__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% le_minus_iff
tff(fact_5478_le__minus__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).
% le_minus_iff
tff(fact_5479_le__minus__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).
% le_minus_iff
tff(fact_5480_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).
% minus_le_iff
tff(fact_5481_minus__le__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).
% minus_le_iff
tff(fact_5482_minus__le__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).
% minus_le_iff
tff(fact_5483_minus__le__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).
% minus_le_iff
tff(fact_5484_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% le_imp_neg_le
tff(fact_5485_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% le_imp_neg_le
tff(fact_5486_le__imp__neg__le,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).
% le_imp_neg_le
tff(fact_5487_le__imp__neg__le,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).
% le_imp_neg_le
tff(fact_5488_less__minus__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).
% less_minus_iff
tff(fact_5489_less__minus__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).
% less_minus_iff
tff(fact_5490_less__minus__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% less_minus_iff
tff(fact_5491_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% less_minus_iff
tff(fact_5492_minus__less__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).
% minus_less_iff
tff(fact_5493_minus__less__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).
% minus_less_iff
tff(fact_5494_minus__less__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).
% minus_less_iff
tff(fact_5495_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).
% minus_less_iff
tff(fact_5496_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).
% verit_negate_coefficient(2)
tff(fact_5497_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).
% verit_negate_coefficient(2)
tff(fact_5498_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% verit_negate_coefficient(2)
tff(fact_5499_verit__negate__coefficient_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% verit_negate_coefficient(2)
tff(fact_5500_numeral__neq__neg__numeral,axiom,
! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).
% numeral_neq_neg_numeral
tff(fact_5501_numeral__neq__neg__numeral,axiom,
! [Mb: num,Nb: num] : aa_num_real(numeral_numeral_real,Mb) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).
% numeral_neq_neg_numeral
tff(fact_5502_numeral__neq__neg__numeral,axiom,
! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).
% numeral_neq_neg_numeral
tff(fact_5503_numeral__neq__neg__numeral,axiom,
! [Mb: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Mb) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).
% numeral_neq_neg_numeral
tff(fact_5504_numeral__neq__neg__numeral,axiom,
! [Mb: num,Nb: num] : numera6620942414471956472nteger(Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).
% numeral_neq_neg_numeral
tff(fact_5505_neg__numeral__neq__numeral,axiom,
! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).
% neg_numeral_neq_numeral
tff(fact_5506_neg__numeral__neq__numeral,axiom,
! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) != aa_num_real(numeral_numeral_real,Nb) ).
% neg_numeral_neq_numeral
tff(fact_5507_neg__numeral__neq__numeral,axiom,
! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).
% neg_numeral_neq_numeral
tff(fact_5508_neg__numeral__neq__numeral,axiom,
! [Mb: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) != aa_num_rat(numeral_numeral_rat,Nb) ).
% neg_numeral_neq_numeral
tff(fact_5509_neg__numeral__neq__numeral,axiom,
! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) != numera6620942414471956472nteger(Nb) ).
% neg_numeral_neq_numeral
tff(fact_5510_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_5511_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_5512_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_5513_minus__mult__commute,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).
% minus_mult_commute
tff(fact_5514_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_5515_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_5516_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_5517_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_5518_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 = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).
% square_eq_iff
tff(fact_5519_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_5520_one__neq__neg__one,axiom,
one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% one_neq_neg_one
tff(fact_5521_one__neq__neg__one,axiom,
one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).
% one_neq_neg_one
tff(fact_5522_one__neq__neg__one,axiom,
one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% one_neq_neg_one
tff(fact_5523_one__neq__neg__one,axiom,
one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% one_neq_neg_one
tff(fact_5524_one__neq__neg__one,axiom,
one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% one_neq_neg_one
tff(fact_5525_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_5526_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_5527_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_5528_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).
% is_num_normalize(8)
tff(fact_5529_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_5530_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(uminus_uminus_int,A3) = 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_5531_group__cancel_Oneg1,axiom,
! [A3: real,K: real,A: real] :
( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
=> ( aa_real_real(uminus_uminus_real,A3) = 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_5532_group__cancel_Oneg1,axiom,
! [A3: complex,K: complex,A: complex] :
( ( A3 = aa_complex_complex(plus_plus_complex(K),A) )
=> ( aa_complex_complex(uminus1482373934393186551omplex,A3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).
% group_cancel.neg1
tff(fact_5533_group__cancel_Oneg1,axiom,
! [A3: rat,K: rat,A: rat] :
( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
=> ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).
% group_cancel.neg1
tff(fact_5534_group__cancel_Oneg1,axiom,
! [A3: code_integer,K: code_integer,A: code_integer] :
( ( A3 = plus_p5714425477246183910nteger(K,A) )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).
% group_cancel.neg1
tff(fact_5535_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_5536_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_5537_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_5538_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).
% add.inverse_distrib_swap
tff(fact_5539_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_5540_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_5541_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_5542_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_5543_minus__diff__commute,axiom,
! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).
% minus_diff_commute
tff(fact_5544_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_5545_case__prodD,axiom,
! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B)))
=> aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B)) ) ).
% case_prodD
tff(fact_5546_case__prodD,axiom,
! [F: fun_num_fun_num_o,A: num,B: num] :
( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B))
=> aa_num_o(aa_num_fun_num_o(F,A),B) ) ).
% case_prodD
tff(fact_5547_case__prodD,axiom,
! [F: fun_nat_fun_num_o,A: nat,B: num] :
( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B))
=> aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).
% case_prodD
tff(fact_5548_case__prodD,axiom,
! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))
=> aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).
% case_prodD
tff(fact_5549_case__prodD,axiom,
! [F: fun_int_fun_int_o,A: int,B: int] :
( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).
% case_prodD
tff(fact_5550_case__prodE,axiom,
! [C: fun_Co480733329897759773un_o_o,P2: produc6271795597528267376eger_o] :
( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2)
=> ~ ! [X4: code_integer,Y3: $o] :
( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
=> ~ aa_o_o(aa_Cod6379977994291289310un_o_o(C,X4),(Y3)) ) ) ).
% case_prodE
tff(fact_5551_case__prodE,axiom,
! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
=> ~ ! [X4: num,Y3: num] :
( ( P2 = product_Pair_num_num(X4,Y3) )
=> ~ aa_num_o(aa_num_fun_num_o(C,X4),Y3) ) ) ).
% case_prodE
tff(fact_5552_case__prodE,axiom,
! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
=> ~ ! [X4: nat,Y3: num] :
( ( P2 = product_Pair_nat_num(X4,Y3) )
=> ~ aa_num_o(aa_nat_fun_num_o(C,X4),Y3) ) ) ).
% case_prodE
tff(fact_5553_case__prodE,axiom,
! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
=> ~ ! [X4: nat,Y3: nat] :
( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
=> ~ aa_nat_o(aa_nat_fun_nat_o(C,X4),Y3) ) ) ).
% case_prodE
tff(fact_5554_case__prodE,axiom,
! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
=> ~ ! [X4: int,Y3: int] :
( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
=> ~ aa_int_o(aa_int_fun_int_o(C,X4),Y3) ) ) ).
% case_prodE
tff(fact_5555_case__prodD_H,axiom,
! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C) ) ).
% case_prodD'
tff(fact_5556_case__prodE_H,axiom,
! [C: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Z)
=> ~ ! [X4: nat,Y3: nat] :
( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
=> ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X4),Y3),Z) ) ) ).
% case_prodE'
tff(fact_5557_zero__less__eq__of__bool,axiom,
! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).
% zero_less_eq_of_bool
tff(fact_5558_zero__less__eq__of__bool,axiom,
! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).
% zero_less_eq_of_bool
tff(fact_5559_zero__less__eq__of__bool,axiom,
! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P))) ).
% zero_less_eq_of_bool
tff(fact_5560_zero__less__eq__of__bool,axiom,
! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).
% zero_less_eq_of_bool
tff(fact_5561_zero__less__eq__of__bool,axiom,
! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).
% zero_less_eq_of_bool
tff(fact_5562_of__bool__less__eq__one,axiom,
! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).
% of_bool_less_eq_one
tff(fact_5563_of__bool__less__eq__one,axiom,
! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).
% of_bool_less_eq_one
tff(fact_5564_of__bool__less__eq__one,axiom,
! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P))),one_one_nat) ).
% of_bool_less_eq_one
tff(fact_5565_of__bool__less__eq__one,axiom,
! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).
% of_bool_less_eq_one
tff(fact_5566_of__bool__less__eq__one,axiom,
! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).
% of_bool_less_eq_one
tff(fact_5567_split__of__bool__asm,axiom,
! [P: fun_complex_o,P2: $o] :
( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
<=> ~ ( ( (P2)
& ~ aa_complex_o(P,one_one_complex) )
| ( ~ (P2)
& ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).
% split_of_bool_asm
tff(fact_5568_split__of__bool__asm,axiom,
! [P: fun_real_o,P2: $o] :
( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
<=> ~ ( ( (P2)
& ~ aa_real_o(P,one_one_real) )
| ( ~ (P2)
& ~ aa_real_o(P,zero_zero_real) ) ) ) ).
% split_of_bool_asm
tff(fact_5569_split__of__bool__asm,axiom,
! [P: fun_rat_o,P2: $o] :
( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
<=> ~ ( ( (P2)
& ~ aa_rat_o(P,one_one_rat) )
| ( ~ (P2)
& ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).
% split_of_bool_asm
tff(fact_5570_split__of__bool__asm,axiom,
! [P: fun_nat_o,P2: $o] :
( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
<=> ~ ( ( (P2)
& ~ aa_nat_o(P,one_one_nat) )
| ( ~ (P2)
& ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).
% split_of_bool_asm
tff(fact_5571_split__of__bool__asm,axiom,
! [P: fun_int_o,P2: $o] :
( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
<=> ~ ( ( (P2)
& ~ aa_int_o(P,one_one_int) )
| ( ~ (P2)
& ~ aa_int_o(P,zero_zero_int) ) ) ) ).
% split_of_bool_asm
tff(fact_5572_split__of__bool__asm,axiom,
! [P: fun_Code_integer_o,P2: $o] :
( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
<=> ~ ( ( (P2)
& ~ aa_Code_integer_o(P,one_one_Code_integer) )
| ( ~ (P2)
& ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).
% split_of_bool_asm
tff(fact_5573_split__of__bool,axiom,
! [P: fun_complex_o,P2: $o] :
( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
<=> ( ( (P2)
=> aa_complex_o(P,one_one_complex) )
& ( ~ (P2)
=> aa_complex_o(P,zero_zero_complex) ) ) ) ).
% split_of_bool
tff(fact_5574_split__of__bool,axiom,
! [P: fun_real_o,P2: $o] :
( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
<=> ( ( (P2)
=> aa_real_o(P,one_one_real) )
& ( ~ (P2)
=> aa_real_o(P,zero_zero_real) ) ) ) ).
% split_of_bool
tff(fact_5575_split__of__bool,axiom,
! [P: fun_rat_o,P2: $o] :
( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
<=> ( ( (P2)
=> aa_rat_o(P,one_one_rat) )
& ( ~ (P2)
=> aa_rat_o(P,zero_zero_rat) ) ) ) ).
% split_of_bool
tff(fact_5576_split__of__bool,axiom,
! [P: fun_nat_o,P2: $o] :
( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
<=> ( ( (P2)
=> aa_nat_o(P,one_one_nat) )
& ( ~ (P2)
=> aa_nat_o(P,zero_zero_nat) ) ) ) ).
% split_of_bool
tff(fact_5577_split__of__bool,axiom,
! [P: fun_int_o,P2: $o] :
( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
<=> ( ( (P2)
=> aa_int_o(P,one_one_int) )
& ( ~ (P2)
=> aa_int_o(P,zero_zero_int) ) ) ) ).
% split_of_bool
tff(fact_5578_split__of__bool,axiom,
! [P: fun_Code_integer_o,P2: $o] :
( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
<=> ( ( (P2)
=> aa_Code_integer_o(P,one_one_Code_integer) )
& ( ~ (P2)
=> aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).
% split_of_bool
tff(fact_5579_of__bool__def,axiom,
! [P2: $o] :
zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).
% of_bool_def
tff(fact_5580_of__bool__def,axiom,
! [P2: $o] :
zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).
% of_bool_def
tff(fact_5581_of__bool__def,axiom,
! [P2: $o] :
zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).
% of_bool_def
tff(fact_5582_of__bool__def,axiom,
! [P2: $o] :
zero_n2687167440665602831ol_nat((P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).
% of_bool_def
tff(fact_5583_of__bool__def,axiom,
! [P2: $o] :
aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).
% of_bool_def
tff(fact_5584_of__bool__def,axiom,
! [P2: $o] :
zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).
% of_bool_def
tff(fact_5585_neg__numeral__le__numeral,axiom,
! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).
% neg_numeral_le_numeral
tff(fact_5586_neg__numeral__le__numeral,axiom,
! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).
% neg_numeral_le_numeral
tff(fact_5587_neg__numeral__le__numeral,axiom,
! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).
% neg_numeral_le_numeral
tff(fact_5588_neg__numeral__le__numeral,axiom,
! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).
% neg_numeral_le_numeral
tff(fact_5589_not__numeral__le__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).
% not_numeral_le_neg_numeral
tff(fact_5590_not__numeral__le__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).
% not_numeral_le_neg_numeral
tff(fact_5591_not__numeral__le__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).
% not_numeral_le_neg_numeral
tff(fact_5592_not__numeral__le__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).
% not_numeral_le_neg_numeral
tff(fact_5593_zero__neq__neg__numeral,axiom,
! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).
% zero_neq_neg_numeral
tff(fact_5594_zero__neq__neg__numeral,axiom,
! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).
% zero_neq_neg_numeral
tff(fact_5595_zero__neq__neg__numeral,axiom,
! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).
% zero_neq_neg_numeral
tff(fact_5596_zero__neq__neg__numeral,axiom,
! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).
% zero_neq_neg_numeral
tff(fact_5597_zero__neq__neg__numeral,axiom,
! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).
% zero_neq_neg_numeral
tff(fact_5598_neg__numeral__less__numeral,axiom,
! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).
% neg_numeral_less_numeral
tff(fact_5599_neg__numeral__less__numeral,axiom,
! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).
% neg_numeral_less_numeral
tff(fact_5600_neg__numeral__less__numeral,axiom,
! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).
% neg_numeral_less_numeral
tff(fact_5601_neg__numeral__less__numeral,axiom,
! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).
% neg_numeral_less_numeral
tff(fact_5602_not__numeral__less__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).
% not_numeral_less_neg_numeral
tff(fact_5603_not__numeral__less__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).
% not_numeral_less_neg_numeral
tff(fact_5604_not__numeral__less__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).
% not_numeral_less_neg_numeral
tff(fact_5605_not__numeral__less__neg__numeral,axiom,
! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).
% not_numeral_less_neg_numeral
tff(fact_5606_le__minus__one__simps_I4_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% le_minus_one_simps(4)
tff(fact_5607_le__minus__one__simps_I4_J,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% le_minus_one_simps(4)
tff(fact_5608_le__minus__one__simps_I4_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% le_minus_one_simps(4)
tff(fact_5609_le__minus__one__simps_I4_J,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).
% le_minus_one_simps(4)
tff(fact_5610_le__minus__one__simps_I2_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).
% le_minus_one_simps(2)
tff(fact_5611_le__minus__one__simps_I2_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).
% le_minus_one_simps(2)
tff(fact_5612_le__minus__one__simps_I2_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).
% le_minus_one_simps(2)
tff(fact_5613_le__minus__one__simps_I2_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).
% le_minus_one_simps(2)
tff(fact_5614_zero__neq__neg__one,axiom,
zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% zero_neq_neg_one
tff(fact_5615_zero__neq__neg__one,axiom,
zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).
% zero_neq_neg_one
tff(fact_5616_zero__neq__neg__one,axiom,
zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% zero_neq_neg_one
tff(fact_5617_zero__neq__neg__one,axiom,
zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% zero_neq_neg_one
tff(fact_5618_zero__neq__neg__one,axiom,
zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% zero_neq_neg_one
tff(fact_5619_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_5620_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_5621_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_5622_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
<=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% add_eq_0_iff
tff(fact_5623_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_5624_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_5625_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_5626_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_5627_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).
% ab_group_add_class.ab_left_minus
tff(fact_5628_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_5629_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_5630_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_5631_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_5632_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).
% add.inverse_unique
tff(fact_5633_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_5634_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_5635_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_5636_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_5637_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_5638_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_5639_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_5640_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_5641_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_5642_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = B )
<=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_5643_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_5644_less__minus__one__simps_I4_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% less_minus_one_simps(4)
tff(fact_5645_less__minus__one__simps_I4_J,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).
% less_minus_one_simps(4)
tff(fact_5646_less__minus__one__simps_I4_J,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% less_minus_one_simps(4)
tff(fact_5647_less__minus__one__simps_I4_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% less_minus_one_simps(4)
tff(fact_5648_less__minus__one__simps_I2_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).
% less_minus_one_simps(2)
tff(fact_5649_less__minus__one__simps_I2_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).
% less_minus_one_simps(2)
tff(fact_5650_less__minus__one__simps_I2_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).
% less_minus_one_simps(2)
tff(fact_5651_less__minus__one__simps_I2_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).
% less_minus_one_simps(2)
tff(fact_5652_numeral__times__minus__swap,axiom,
! [W: num,Xb: 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,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).
% numeral_times_minus_swap
tff(fact_5653_numeral__times__minus__swap,axiom,
! [W: num,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).
% numeral_times_minus_swap
tff(fact_5654_numeral__times__minus__swap,axiom,
! [W: num,Xb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).
% numeral_times_minus_swap
tff(fact_5655_numeral__times__minus__swap,axiom,
! [W: num,Xb: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xb)) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).
% numeral_times_minus_swap
tff(fact_5656_numeral__times__minus__swap,axiom,
! [W: num,Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).
% numeral_times_minus_swap
tff(fact_5657_numeral__neq__neg__one,axiom,
! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).
% numeral_neq_neg_one
tff(fact_5658_numeral__neq__neg__one,axiom,
! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).
% numeral_neq_neg_one
tff(fact_5659_numeral__neq__neg__one,axiom,
! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% numeral_neq_neg_one
tff(fact_5660_numeral__neq__neg__one,axiom,
! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% numeral_neq_neg_one
tff(fact_5661_numeral__neq__neg__one,axiom,
! [Nb: num] : numera6620942414471956472nteger(Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% numeral_neq_neg_one
tff(fact_5662_one__neq__neg__numeral,axiom,
! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).
% one_neq_neg_numeral
tff(fact_5663_one__neq__neg__numeral,axiom,
! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).
% one_neq_neg_numeral
tff(fact_5664_one__neq__neg__numeral,axiom,
! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).
% one_neq_neg_numeral
tff(fact_5665_one__neq__neg__numeral,axiom,
! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).
% one_neq_neg_numeral
tff(fact_5666_one__neq__neg__numeral,axiom,
! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).
% one_neq_neg_numeral
tff(fact_5667_square__eq__1__iff,axiom,
! [Xb: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
<=> ( ( Xb = one_one_int )
| ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% square_eq_1_iff
tff(fact_5668_square__eq__1__iff,axiom,
! [Xb: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb) = one_one_real )
<=> ( ( Xb = one_one_real )
| ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).
% square_eq_1_iff
tff(fact_5669_square__eq__1__iff,axiom,
! [Xb: complex] :
( ( aa_complex_complex(times_times_complex(Xb),Xb) = one_one_complex )
<=> ( ( Xb = one_one_complex )
| ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).
% square_eq_1_iff
tff(fact_5670_square__eq__1__iff,axiom,
! [Xb: rat] :
( ( aa_rat_rat(times_times_rat(Xb),Xb) = one_one_rat )
<=> ( ( Xb = one_one_rat )
| ( Xb = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).
% square_eq_1_iff
tff(fact_5671_square__eq__1__iff,axiom,
! [Xb: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xb) = one_one_Code_integer )
<=> ( ( Xb = one_one_Code_integer )
| ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% square_eq_1_iff
tff(fact_5672_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3 = 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),B3) = 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_5673_group__cancel_Osub2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B3) = 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_5674_group__cancel_Osub2,axiom,
! [B3: complex,K: complex,B: complex,A: complex] :
( ( B3 = aa_complex_complex(plus_plus_complex(K),B) )
=> ( aa_complex_complex(minus_minus_complex(A),B3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).
% group_cancel.sub2
tff(fact_5675_group__cancel_Osub2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3 = aa_rat_rat(plus_plus_rat(K),B) )
=> ( aa_rat_rat(minus_minus_rat(A),B3) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).
% group_cancel.sub2
tff(fact_5676_group__cancel_Osub2,axiom,
! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B3 = plus_p5714425477246183910nteger(K,B) )
=> ( minus_8373710615458151222nteger(A,B3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).
% group_cancel.sub2
tff(fact_5677_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_5678_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_5679_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_5680_diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).
% diff_conv_add_uminus
tff(fact_5681_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_5682_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_5683_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_5684_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_5685_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5686_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_5687_replicate__eqI,axiom,
! [Xs: list_P6011104703257516679at_nat,Nb: nat,Xb: product_prod_nat_nat] :
( ( size_s5460976970255530739at_nat(Xs) = Nb )
=> ( ! [Y3: product_prod_nat_nat] :
( member8440522571783428010at_nat(Y3,set_Pr5648618587558075414at_nat(Xs))
=> ( Y3 = Xb ) )
=> ( Xs = replic4235873036481779905at_nat(Nb,Xb) ) ) ) ).
% replicate_eqI
tff(fact_5688_replicate__eqI,axiom,
! [Xs: list_complex,Nb: nat,Xb: complex] :
( ( size_s3451745648224563538omplex(Xs) = Nb )
=> ( ! [Y3: complex] :
( aa_set_complex_o(member_complex(Y3),set_complex2(Xs))
=> ( Y3 = Xb ) )
=> ( Xs = replicate_complex(Nb,Xb) ) ) ) ).
% replicate_eqI
tff(fact_5689_replicate__eqI,axiom,
! [Xs: list_real,Nb: nat,Xb: real] :
( ( size_size_list_real(Xs) = Nb )
=> ( ! [Y3: real] :
( aa_set_real_o(member_real(Y3),set_real2(Xs))
=> ( Y3 = Xb ) )
=> ( Xs = replicate_real(Nb,Xb) ) ) ) ).
% replicate_eqI
tff(fact_5690_replicate__eqI,axiom,
! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
=> ( ! [Y3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y3),set_VEBT_VEBT2(Xs))
=> ( Y3 = Xb ) )
=> ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).
% replicate_eqI
tff(fact_5691_replicate__eqI,axiom,
! [Xs: list_o,Nb: nat,Xb: $o] :
( ( size_size_list_o(Xs) = Nb )
=> ( ! [Y3: $o] :
( member_o((Y3),set_o2(Xs))
=> ( (Y3)
<=> (Xb) ) )
=> ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).
% replicate_eqI
tff(fact_5692_replicate__eqI,axiom,
! [Xs: list_nat,Nb: nat,Xb: nat] :
( ( size_size_list_nat(Xs) = Nb )
=> ( ! [Y3: nat] :
( aa_set_nat_o(member_nat(Y3),set_nat2(Xs))
=> ( Y3 = Xb ) )
=> ( Xs = replicate_nat(Nb,Xb) ) ) ) ).
% replicate_eqI
tff(fact_5693_replicate__eqI,axiom,
! [Xs: list_int,Nb: nat,Xb: int] :
( ( size_size_list_int(Xs) = Nb )
=> ( ! [Y3: int] :
( aa_set_int_o(member_int(Y3),set_int2(Xs))
=> ( Y3 = Xb ) )
=> ( Xs = replicate_int(Nb,Xb) ) ) ) ).
% replicate_eqI
tff(fact_5694_replicate__length__same,axiom,
! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Xs))
=> ( X4 = Xb ) )
=> ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).
% replicate_length_same
tff(fact_5695_replicate__length__same,axiom,
! [Xs: list_o,Xb: $o] :
( ! [X4: $o] :
( member_o((X4),set_o2(Xs))
=> ( (X4)
<=> (Xb) ) )
=> ( replicate_o(size_size_list_o(Xs),(Xb)) = Xs ) ) ).
% replicate_length_same
tff(fact_5696_replicate__length__same,axiom,
! [Xs: list_nat,Xb: nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
=> ( X4 = Xb ) )
=> ( replicate_nat(size_size_list_nat(Xs),Xb) = Xs ) ) ).
% replicate_length_same
tff(fact_5697_replicate__length__same,axiom,
! [Xs: list_int,Xb: int] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),set_int2(Xs))
=> ( X4 = Xb ) )
=> ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).
% replicate_length_same
tff(fact_5698_real__minus__mult__self__le,axiom,
! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(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,Xb),Xb)) ).
% real_minus_mult_self_le
tff(fact_5699_zmult__eq__1__iff,axiom,
! [Mb: int,Nb: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
<=> ( ( ( Mb = one_one_int )
& ( Nb = one_one_int ) )
| ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
& ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).
% zmult_eq_1_iff
tff(fact_5700_pos__zmult__eq__1__iff__lemma,axiom,
! [Mb: int,Nb: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
=> ( ( Mb = one_one_int )
| ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
tff(fact_5701_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_5702_not__zero__le__neg__numeral,axiom,
! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).
% not_zero_le_neg_numeral
tff(fact_5703_not__zero__le__neg__numeral,axiom,
! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).
% not_zero_le_neg_numeral
tff(fact_5704_not__zero__le__neg__numeral,axiom,
! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).
% not_zero_le_neg_numeral
tff(fact_5705_not__zero__le__neg__numeral,axiom,
! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).
% not_zero_le_neg_numeral
tff(fact_5706_neg__numeral__le__zero,axiom,
! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).
% neg_numeral_le_zero
tff(fact_5707_neg__numeral__le__zero,axiom,
! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).
% neg_numeral_le_zero
tff(fact_5708_neg__numeral__le__zero,axiom,
! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).
% neg_numeral_le_zero
tff(fact_5709_neg__numeral__le__zero,axiom,
! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).
% neg_numeral_le_zero
tff(fact_5710_not__zero__less__neg__numeral,axiom,
! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).
% not_zero_less_neg_numeral
tff(fact_5711_not__zero__less__neg__numeral,axiom,
! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).
% not_zero_less_neg_numeral
tff(fact_5712_not__zero__less__neg__numeral,axiom,
! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).
% not_zero_less_neg_numeral
tff(fact_5713_not__zero__less__neg__numeral,axiom,
! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).
% not_zero_less_neg_numeral
tff(fact_5714_neg__numeral__less__zero,axiom,
! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).
% neg_numeral_less_zero
tff(fact_5715_neg__numeral__less__zero,axiom,
! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).
% neg_numeral_less_zero
tff(fact_5716_neg__numeral__less__zero,axiom,
! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).
% neg_numeral_less_zero
tff(fact_5717_neg__numeral__less__zero,axiom,
! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).
% neg_numeral_less_zero
tff(fact_5718_le__minus__one__simps_I3_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% le_minus_one_simps(3)
tff(fact_5719_le__minus__one__simps_I3_J,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% le_minus_one_simps(3)
tff(fact_5720_le__minus__one__simps_I3_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% le_minus_one_simps(3)
tff(fact_5721_le__minus__one__simps_I3_J,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).
% le_minus_one_simps(3)
tff(fact_5722_le__minus__one__simps_I1_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).
% le_minus_one_simps(1)
tff(fact_5723_le__minus__one__simps_I1_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).
% le_minus_one_simps(1)
tff(fact_5724_le__minus__one__simps_I1_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).
% le_minus_one_simps(1)
tff(fact_5725_le__minus__one__simps_I1_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).
% le_minus_one_simps(1)
tff(fact_5726_less__minus__one__simps_I1_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).
% less_minus_one_simps(1)
tff(fact_5727_less__minus__one__simps_I1_J,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).
% less_minus_one_simps(1)
tff(fact_5728_less__minus__one__simps_I1_J,axiom,
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).
% less_minus_one_simps(1)
tff(fact_5729_less__minus__one__simps_I1_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).
% less_minus_one_simps(1)
tff(fact_5730_less__minus__one__simps_I3_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% less_minus_one_simps(3)
tff(fact_5731_less__minus__one__simps_I3_J,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).
% less_minus_one_simps(3)
tff(fact_5732_less__minus__one__simps_I3_J,axiom,
~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% less_minus_one_simps(3)
tff(fact_5733_less__minus__one__simps_I3_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% less_minus_one_simps(3)
tff(fact_5734_neg__numeral__le__one,axiom,
! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).
% neg_numeral_le_one
tff(fact_5735_neg__numeral__le__one,axiom,
! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).
% neg_numeral_le_one
tff(fact_5736_neg__numeral__le__one,axiom,
! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).
% neg_numeral_le_one
tff(fact_5737_neg__numeral__le__one,axiom,
! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).
% neg_numeral_le_one
tff(fact_5738_neg__one__le__numeral,axiom,
! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).
% neg_one_le_numeral
tff(fact_5739_neg__one__le__numeral,axiom,
! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).
% neg_one_le_numeral
tff(fact_5740_neg__one__le__numeral,axiom,
! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).
% neg_one_le_numeral
tff(fact_5741_neg__one__le__numeral,axiom,
! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).
% neg_one_le_numeral
tff(fact_5742_neg__numeral__le__neg__one,axiom,
! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% neg_numeral_le_neg_one
tff(fact_5743_neg__numeral__le__neg__one,axiom,
! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% neg_numeral_le_neg_one
tff(fact_5744_neg__numeral__le__neg__one,axiom,
! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).
% neg_numeral_le_neg_one
tff(fact_5745_neg__numeral__le__neg__one,axiom,
! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).
% neg_numeral_le_neg_one
tff(fact_5746_not__numeral__le__neg__one,axiom,
! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% not_numeral_le_neg_one
tff(fact_5747_not__numeral__le__neg__one,axiom,
! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% not_numeral_le_neg_one
tff(fact_5748_not__numeral__le__neg__one,axiom,
! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).
% not_numeral_le_neg_one
tff(fact_5749_not__numeral__le__neg__one,axiom,
! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).
% not_numeral_le_neg_one
tff(fact_5750_not__one__le__neg__numeral,axiom,
! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).
% not_one_le_neg_numeral
tff(fact_5751_not__one__le__neg__numeral,axiom,
! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).
% not_one_le_neg_numeral
tff(fact_5752_not__one__le__neg__numeral,axiom,
! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).
% not_one_le_neg_numeral
tff(fact_5753_not__one__le__neg__numeral,axiom,
! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).
% not_one_le_neg_numeral
tff(fact_5754_not__neg__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).
% not_neg_one_less_neg_numeral
tff(fact_5755_not__neg__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).
% not_neg_one_less_neg_numeral
tff(fact_5756_not__neg__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).
% not_neg_one_less_neg_numeral
tff(fact_5757_not__neg__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).
% not_neg_one_less_neg_numeral
tff(fact_5758_not__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).
% not_one_less_neg_numeral
tff(fact_5759_not__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).
% not_one_less_neg_numeral
tff(fact_5760_not__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).
% not_one_less_neg_numeral
tff(fact_5761_not__one__less__neg__numeral,axiom,
! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).
% not_one_less_neg_numeral
tff(fact_5762_not__numeral__less__neg__one,axiom,
! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).
% not_numeral_less_neg_one
tff(fact_5763_not__numeral__less__neg__one,axiom,
! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).
% not_numeral_less_neg_one
tff(fact_5764_not__numeral__less__neg__one,axiom,
! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% not_numeral_less_neg_one
tff(fact_5765_not__numeral__less__neg__one,axiom,
! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% not_numeral_less_neg_one
tff(fact_5766_neg__one__less__numeral,axiom,
! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).
% neg_one_less_numeral
tff(fact_5767_neg__one__less__numeral,axiom,
! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).
% neg_one_less_numeral
tff(fact_5768_neg__one__less__numeral,axiom,
! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).
% neg_one_less_numeral
tff(fact_5769_neg__one__less__numeral,axiom,
! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).
% neg_one_less_numeral
tff(fact_5770_neg__numeral__less__one,axiom,
! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).
% neg_numeral_less_one
tff(fact_5771_neg__numeral__less__one,axiom,
! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).
% neg_numeral_less_one
tff(fact_5772_neg__numeral__less__one,axiom,
! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).
% neg_numeral_less_one
tff(fact_5773_neg__numeral__less__one,axiom,
! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).
% neg_numeral_less_one
tff(fact_5774_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)) )
<=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).
% eq_minus_divide_eq
tff(fact_5775_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
<=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).
% eq_minus_divide_eq
tff(fact_5776_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).
% eq_minus_divide_eq
tff(fact_5777_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 )
<=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).
% minus_divide_eq_eq
tff(fact_5778_minus__divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
<=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).
% minus_divide_eq_eq
tff(fact_5779_minus__divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
<=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).
% minus_divide_eq_eq
tff(fact_5780_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_5781_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_5782_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( B != zero_zero_rat )
=> ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
<=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_5783_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_5784_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_5785_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
<=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_5786_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_5787_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_5788_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
<=> ( ( B != zero_zero_rat )
& ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_5789_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_5790_mult__1s__ring__1_I1_J,axiom,
! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).
% mult_1s_ring_1(1)
tff(fact_5791_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_5792_mult__1s__ring__1_I1_J,axiom,
! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).
% mult_1s_ring_1(1)
tff(fact_5793_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_5794_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_5795_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,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).
% mult_1s_ring_1(2)
tff(fact_5796_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_5797_mult__1s__ring__1_I2_J,axiom,
! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).
% mult_1s_ring_1(2)
tff(fact_5798_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_5799_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_5800_uminus__numeral__One,axiom,
aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).
% uminus_numeral_One
tff(fact_5801_uminus__numeral__One,axiom,
aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% uminus_numeral_One
tff(fact_5802_uminus__numeral__One,axiom,
aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% uminus_numeral_One
tff(fact_5803_uminus__numeral__One,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% uminus_numeral_One
tff(fact_5804_power__minus,axiom,
! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = 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)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).
% power_minus
tff(fact_5805_power__minus,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = 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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% power_minus
tff(fact_5806_power__minus,axiom,
! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(A),Nb)) ).
% power_minus
tff(fact_5807_power__minus,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).
% power_minus
tff(fact_5808_power__minus,axiom,
! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).
% power_minus
tff(fact_5809_power__minus__Bit0,axiom,
! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5810_power__minus__Bit0,axiom,
! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5811_power__minus__Bit0,axiom,
! [Xb: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5812_power__minus__Bit0,axiom,
! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5813_power__minus__Bit0,axiom,
! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5814_power__minus__Bit1,axiom,
! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5815_power__minus__Bit1,axiom,
! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5816_power__minus__Bit1,axiom,
! [Xb: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5817_power__minus__Bit1,axiom,
! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5818_power__minus__Bit1,axiom,
! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5819_real__0__less__add__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).
% real_0_less_add_iff
tff(fact_5820_real__add__less__0__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).
% real_add_less_0_iff
tff(fact_5821_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_5822_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).
% less_minus_divide_eq
tff(fact_5823_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).
% less_minus_divide_eq
tff(fact_5824_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).
% minus_divide_less_eq
tff(fact_5825_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).
% minus_divide_less_eq
tff(fact_5826_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> aa_real_o(aa_real_fun_real_o(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_5827_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% neg_less_minus_divide_eq
tff(fact_5828_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
<=> aa_real_o(aa_real_fun_real_o(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_5829_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% neg_minus_divide_less_eq
tff(fact_5830_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> aa_real_o(aa_real_fun_real_o(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_5831_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% pos_less_minus_divide_eq
tff(fact_5832_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
<=> aa_real_o(aa_real_fun_real_o(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_5833_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% pos_minus_divide_less_eq
tff(fact_5834_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,aa_num_real(numeral_numeral_real,W)) )
<=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).
% divide_eq_eq_numeral(2)
tff(fact_5835_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)) )
<=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).
% divide_eq_eq_numeral(2)
tff(fact_5836_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
<=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).
% divide_eq_eq_numeral(2)
tff(fact_5837_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).
% eq_divide_eq_numeral(2)
tff(fact_5838_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) )
<=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).
% eq_divide_eq_numeral(2)
tff(fact_5839_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).
% eq_divide_eq_numeral(2)
tff(fact_5840_add__divide__eq__if__simps_I3_J,axiom,
! [A: real,Z: real,B: 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) = $ite(Z = zero_zero_real,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_5841_add__divide__eq__if__simps_I3_J,axiom,
! [A: complex,Z: complex,B: complex] :
aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,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_5842_add__divide__eq__if__simps_I3_J,axiom,
! [A: rat,Z: rat,B: rat] :
aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).
% add_divide_eq_if_simps(3)
tff(fact_5843_minus__divide__add__eq__iff,axiom,
! [Z: real,Xb: 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,Xb),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,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_5844_minus__divide__add__eq__iff,axiom,
! [Z: complex,Xb: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_5845_minus__divide__add__eq__iff,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_5846_subset__decode__imp__le,axiom,
! [Mb: nat,Nb: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,Mb)),aa_nat_set_nat(nat_set_decode,Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% subset_decode_imp_le
tff(fact_5847_minus__divide__diff__eq__iff,axiom,
! [Z: real,Xb: 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,Xb),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,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_5848_minus__divide__diff__eq__iff,axiom,
! [Z: complex,Xb: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_5849_minus__divide__diff__eq__iff,axiom,
! [Z: rat,Xb: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_5850_add__divide__eq__if__simps_I5_J,axiom,
! [A: real,Z: real,B: real] :
aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,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_5851_add__divide__eq__if__simps_I5_J,axiom,
! [A: complex,Z: complex,B: complex] :
aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,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_5852_add__divide__eq__if__simps_I5_J,axiom,
! [A: rat,Z: rat,B: rat] :
aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).
% add_divide_eq_if_simps(5)
tff(fact_5853_add__divide__eq__if__simps_I6_J,axiom,
! [A: real,Z: real,B: 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) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,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_5854_add__divide__eq__if__simps_I6_J,axiom,
! [A: complex,Z: complex,B: complex] :
aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,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_5855_add__divide__eq__if__simps_I6_J,axiom,
! [A: rat,Z: rat,B: rat] :
aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).
% add_divide_eq_if_simps(6)
tff(fact_5856_even__minus,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).
% even_minus
tff(fact_5857_even__minus,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).
% even_minus
tff(fact_5858_power2__eq__iff,axiom,
! [Xb: int,Y: int] :
( ( aa_nat_int(power_power_int(Xb),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))) )
<=> ( ( Xb = Y )
| ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5859_power2__eq__iff,axiom,
! [Xb: real,Y: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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))) )
<=> ( ( Xb = Y )
| ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5860_power2__eq__iff,axiom,
! [Xb: complex,Y: complex] :
( ( aa_nat_complex(power_power_complex(Xb),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))) )
<=> ( ( Xb = Y )
| ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5861_power2__eq__iff,axiom,
! [Xb: rat,Y: rat] :
( ( aa_nat_rat(power_power_rat(Xb),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))) )
<=> ( ( Xb = Y )
| ( Xb = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5862_power2__eq__iff,axiom,
! [Xb: code_integer,Y: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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))) )
<=> ( ( Xb = Y )
| ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5863_uminus__power__if,axiom,
! [A: int,Nb: nat] :
aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(power_power_int(A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb))) ).
% uminus_power_if
tff(fact_5864_uminus__power__if,axiom,
! [A: real,Nb: nat] :
aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ).
% uminus_power_if
tff(fact_5865_uminus__power__if,axiom,
! [A: complex,Nb: nat] :
aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(power_power_complex(A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb))) ).
% uminus_power_if
tff(fact_5866_uminus__power__if,axiom,
! [A: rat,Nb: nat] :
aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb))) ).
% uminus_power_if
tff(fact_5867_uminus__power__if,axiom,
! [A: code_integer,Nb: nat] :
aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).
% uminus_power_if
tff(fact_5868_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(B3),Nb)),aa_int_int(divide_divide_int(A3),Nb)) ) ) ).
% verit_less_mono_div_int2
tff(fact_5869_div__eq__minus1,axiom,
! [B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_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_5870_of__bool__odd__eq__mod__2,axiom,
! [A: nat] : zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(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_5871_of__bool__odd__eq__mod__2,axiom,
! [A: int] : aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_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_5872_of__bool__odd__eq__mod__2,axiom,
! [A: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5873_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% pos_minus_divide_le_eq
tff(fact_5874_pos__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
<=> aa_real_o(aa_real_fun_real_o(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_5875_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% pos_le_minus_divide_eq
tff(fact_5876_pos__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> aa_real_o(aa_real_fun_real_o(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_5877_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% neg_minus_divide_le_eq
tff(fact_5878_neg__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
<=> aa_real_o(aa_real_fun_real_o(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_5879_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).
% neg_le_minus_divide_eq
tff(fact_5880_neg__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> aa_real_o(aa_real_fun_real_o(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_5881_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).
% minus_divide_le_eq
tff(fact_5882_minus__divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).
% minus_divide_le_eq
tff(fact_5883_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).
% le_minus_divide_eq
tff(fact_5884_le__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).
% le_minus_divide_eq
tff(fact_5885_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_5886_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_5887_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_5888_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_5889_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_5890_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_5891_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_5892_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 = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).
% power2_eq_1_iff
tff(fact_5893_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_5894_minus__one__power__iff,axiom,
! [Nb: nat] :
aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).
% minus_one_power_iff
tff(fact_5895_minus__one__power__iff,axiom,
! [Nb: nat] :
aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).
% minus_one_power_iff
tff(fact_5896_minus__one__power__iff,axiom,
! [Nb: nat] :
aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).
% minus_one_power_iff
tff(fact_5897_minus__one__power__iff,axiom,
! [Nb: nat] :
aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% minus_one_power_iff
tff(fact_5898_minus__one__power__iff,axiom,
! [Nb: nat] :
aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% minus_one_power_iff
tff(fact_5899_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( 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,Nb),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,Nb),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5900_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( 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,Nb),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,Nb),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5901_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5902_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5903_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5904_realpow__square__minus__le,axiom,
! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% realpow_square_minus_le
tff(fact_5905_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ).
% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5906_signed__take__bit__int__less__eq__self__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),K)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K) ) ).
% signed_take_bit_int_less_eq_self_iff
tff(fact_5907_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).
% signed_take_bit_int_greater_self_iff
tff(fact_5908_zmod__minus1,axiom,
! [B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_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_5909_bits__induct,axiom,
! [P: fun_nat_o,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 )
=> aa_nat_o(P,A4) )
=> ( ! [A4: nat,B4: $o] :
( aa_nat_o(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 )
=> aa_nat_o(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))) ) )
=> aa_nat_o(P,A) ) ) ).
% bits_induct
tff(fact_5910_bits__induct,axiom,
! [P: fun_int_o,A: int] :
( ! [A4: int] :
( ( aa_int_int(divide_divide_int(A4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
=> aa_int_o(P,A4) )
=> ( ! [A4: int,B4: $o] :
( aa_int_o(P,A4)
=> ( ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_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 )
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_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_int_o(P,A) ) ) ).
% bits_induct
tff(fact_5911_bits__induct,axiom,
! [P: fun_Code_integer_o,A: code_integer] :
( ! [A4: code_integer] :
( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
=> aa_Code_integer_o(P,A4) )
=> ( ! [A4: code_integer,B4: $o] :
( aa_Code_integer_o(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 )
=> aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4))) ) )
=> aa_Code_integer_o(P,A) ) ) ).
% bits_induct
tff(fact_5912_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_5913_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_5914_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
<=> $ite(
aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_5915_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
<=> $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_5916_square__le__1,axiom,
! [Xb: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).
% square_le_1
tff(fact_5917_square__le__1,axiom,
! [Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).
% square_le_1
tff(fact_5918_square__le__1,axiom,
! [Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).
% square_le_1
tff(fact_5919_square__le__1,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).
% square_le_1
tff(fact_5920_minus__power__mult__self,axiom,
! [A: int,Nb: 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)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = 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))),Nb)) ).
% minus_power_mult_self
tff(fact_5921_minus__power__mult__self,axiom,
! [A: real,Nb: 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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = 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))),Nb)) ).
% minus_power_mult_self
tff(fact_5922_minus__power__mult__self,axiom,
! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = 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))),Nb)) ).
% minus_power_mult_self
tff(fact_5923_minus__power__mult__self,axiom,
! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = 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))),Nb)) ).
% minus_power_mult_self
tff(fact_5924_minus__power__mult__self,axiom,
! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = 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))),Nb)) ).
% minus_power_mult_self
tff(fact_5925_signed__take__bit__int__eq__self,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K ) ) ) ).
% signed_take_bit_int_eq_self
tff(fact_5926_signed__take__bit__int__eq__self__iff,axiom,
! [Nb: nat,K: int] :
( ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).
% signed_take_bit_int_eq_self_iff
tff(fact_5927_minus__1__div__exp__eq__int,axiom,
! [Nb: nat] : aa_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))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% minus_1_div_exp_eq_int
tff(fact_5928_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
=> ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% div_pos_neg_trivial
tff(fact_5929_exp__mod__exp,axiom,
! [Mb: nat,Nb: 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))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).
% exp_mod_exp
tff(fact_5930_exp__mod__exp,axiom,
! [Mb: nat,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).
% exp_mod_exp
tff(fact_5931_exp__mod__exp,axiom,
! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) ).
% exp_mod_exp
tff(fact_5932_divmod__nat__def,axiom,
! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).
% divmod_nat_def
tff(fact_5933_power__minus1__odd,axiom,
! [Nb: 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))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).
% power_minus1_odd
tff(fact_5934_power__minus1__odd,axiom,
! [Nb: 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))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).
% power_minus1_odd
tff(fact_5935_power__minus1__odd,axiom,
! [Nb: 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))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% power_minus1_odd
tff(fact_5936_power__minus1__odd,axiom,
! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_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))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% power_minus1_odd
tff(fact_5937_power__minus1__odd,axiom,
! [Nb: 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))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% power_minus1_odd
tff(fact_5938_int__bit__induct,axiom,
! [P: fun_int_o,K: int] :
( aa_int_o(P,zero_zero_int)
=> ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
=> ( ! [K3: int] :
( aa_int_o(P,K3)
=> ( ( K3 != zero_zero_int )
=> aa_int_o(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] :
( aa_int_o(P,K3)
=> ( ( K3 != aa_int_int(uminus_uminus_int,one_one_int) )
=> aa_int_o(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))))) ) )
=> aa_int_o(P,K) ) ) ) ) ).
% int_bit_induct
tff(fact_5939_signed__take__bit__int__greater__eq,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
=> aa_int_o(aa_int_fun_int_o(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,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).
% signed_take_bit_int_greater_eq
tff(fact_5940_set__decode__def,axiom,
! [Xb: nat] : aa_nat_set_nat(nat_set_decode,Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ef,Xb)) ).
% set_decode_def
tff(fact_5941_exp__div__exp__eq,axiom,
! [Mb: nat,Nb: 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))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
zero_n2687167440665602831ol_nat(( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).
% exp_div_exp_eq
tff(fact_5942_exp__div__exp__eq,axiom,
! [Mb: nat,Nb: nat] :
aa_int_int(divide_divide_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).
% exp_div_exp_eq
tff(fact_5943_exp__div__exp__eq,axiom,
! [Mb: nat,Nb: nat] :
divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).
% exp_div_exp_eq
tff(fact_5944_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
$let(
half: nat,
half:= 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_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))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
$let(
half: nat,
half:= 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_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,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).
% vebt_buildup.simps(3)
tff(fact_5945_one__div__minus__numeral,axiom,
! [Nb: num] : aa_int_int(divide_divide_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).
% one_div_minus_numeral
tff(fact_5946_minus__one__div__numeral,axiom,
! [Nb: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).
% minus_one_div_numeral
tff(fact_5947_compl__le__compl__iff,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),uminus5710092332889474511et_nat(Y))
<=> aa_set_nat_o(ord_less_eq_set_nat(Y),Xb) ) ).
% compl_le_compl_iff
tff(fact_5948_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_5949_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5950_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_5951_dbl__dec__simps_I4_J,axiom,
aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5952_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_5953_of__int__code__if,axiom,
! [K: int] :
aa_int_int(ring_1_of_int_int,K) = $ite(
K = zero_zero_int,
zero_zero_int,
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
$let(
l: int,
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(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
$ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).
% of_int_code_if
tff(fact_5954_of__int__code__if,axiom,
! [K: int] :
aa_int_real(ring_1_of_int_real,K) = $ite(
K = zero_zero_int,
zero_zero_real,
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
$let(
l: real,
l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
$ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).
% of_int_code_if
tff(fact_5955_of__int__code__if,axiom,
! [K: int] :
aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
K = zero_zero_int,
zero_zero_complex,
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
$let(
l: complex,
l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
$ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).
% of_int_code_if
tff(fact_5956_of__int__code__if,axiom,
! [K: int] :
aa_int_rat(ring_1_of_int_rat,K) = $ite(
K = zero_zero_int,
zero_zero_rat,
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
$let(
l: rat,
l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
$ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).
% of_int_code_if
tff(fact_5957_of__int__code__if,axiom,
! [K: int] :
aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(
K = zero_zero_int,
zero_z3403309356797280102nteger,
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))),
$let(
l: code_integer,
l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
$ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).
% of_int_code_if
tff(fact_5958_split__part,axiom,
! [P: $o,Q: fun_int_fun_int_o,X3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_eg((P)),Q)),X3)
<=> ( (P)
& aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X3) ) ) ).
% split_part
tff(fact_5959_dbl__dec__simps_I3_J,axiom,
aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).
% dbl_dec_simps(3)
tff(fact_5960_dbl__dec__simps_I3_J,axiom,
aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).
% dbl_dec_simps(3)
tff(fact_5961_dbl__dec__simps_I3_J,axiom,
aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).
% dbl_dec_simps(3)
tff(fact_5962_dbl__dec__simps_I3_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).
% dbl_dec_simps(3)
tff(fact_5963_of__int__numeral,axiom,
! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).
% of_int_numeral
tff(fact_5964_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_5965_of__int__numeral,axiom,
! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).
% of_int_numeral
tff(fact_5966_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_5967_of__int__eq__numeral__iff,axiom,
! [Z: int,Nb: num] :
( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
<=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5968_of__int__eq__numeral__iff,axiom,
! [Z: int,Nb: num] :
( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
<=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5969_of__int__eq__numeral__iff,axiom,
! [Z: int,Nb: num] :
( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
<=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5970_of__int__eq__numeral__iff,axiom,
! [Z: int,Nb: num] :
( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
<=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5971_of__int__le__iff,axiom,
! [W: int,Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).
% of_int_le_iff
tff(fact_5972_of__int__le__iff,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).
% of_int_le_iff
tff(fact_5973_of__int__le__iff,axiom,
! [W: int,Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).
% of_int_le_iff
tff(fact_5974_of__int__less__iff,axiom,
! [W: int,Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).
% of_int_less_iff
tff(fact_5975_of__int__less__iff,axiom,
! [W: int,Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).
% of_int_less_iff
tff(fact_5976_of__int__less__iff,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).
% of_int_less_iff
tff(fact_5977_of__int__1,axiom,
aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).
% of_int_1
tff(fact_5978_of__int__1,axiom,
aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).
% of_int_1
tff(fact_5979_of__int__1,axiom,
aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).
% of_int_1
tff(fact_5980_of__int__1,axiom,
aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).
% of_int_1
tff(fact_5981_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_5982_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_5983_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_5984_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_5985_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_5986_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_5987_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_5988_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_5989_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_5990_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)) = aa_rat_rat(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_5991_of__int__power,axiom,
! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).
% of_int_power
tff(fact_5992_of__int__power,axiom,
! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).
% of_int_power
tff(fact_5993_of__int__power,axiom,
! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).
% of_int_power
tff(fact_5994_of__int__power,axiom,
! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).
% of_int_power
tff(fact_5995_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: 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,Xb) )
<=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5996_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: 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,Xb) )
<=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5997_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: 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,Xb) )
<=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5998_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
<=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5999_of__int__power__eq__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
<=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_6000_of__int__power__eq__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
<=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_6001_of__int__power__eq__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
<=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_6002_of__int__power__eq__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
<=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_6003_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_6004_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_6005_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_6006_dbl__dec__simps_I2_J,axiom,
aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% dbl_dec_simps(2)
tff(fact_6007_dbl__dec__simps_I2_J,axiom,
neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_dec_simps(2)
tff(fact_6008_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_6009_dbl__inc__simps_I1_J,axiom,
! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).
% dbl_inc_simps(1)
tff(fact_6010_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_6011_dbl__inc__simps_I1_J,axiom,
! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).
% dbl_inc_simps(1)
tff(fact_6012_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_6013_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_6014_dbl__dec__simps_I1_J,axiom,
! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).
% dbl_dec_simps(1)
tff(fact_6015_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_6016_dbl__dec__simps_I1_J,axiom,
! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).
% dbl_dec_simps(1)
tff(fact_6017_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_6018_of__int__0__le__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).
% of_int_0_le_iff
tff(fact_6019_of__int__0__le__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).
% of_int_0_le_iff
tff(fact_6020_of__int__0__le__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).
% of_int_0_le_iff
tff(fact_6021_of__int__le__0__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).
% of_int_le_0_iff
tff(fact_6022_of__int__le__0__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).
% of_int_le_0_iff
tff(fact_6023_of__int__le__0__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).
% of_int_le_0_iff
tff(fact_6024_of__int__0__less__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).
% of_int_0_less_iff
tff(fact_6025_of__int__0__less__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).
% of_int_0_less_iff
tff(fact_6026_of__int__0__less__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).
% of_int_0_less_iff
tff(fact_6027_of__int__less__0__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).
% of_int_less_0_iff
tff(fact_6028_of__int__less__0__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).
% of_int_less_0_iff
tff(fact_6029_of__int__less__0__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).
% of_int_less_0_iff
tff(fact_6030_of__int__le__numeral__iff,axiom,
! [Z: int,Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).
% of_int_le_numeral_iff
tff(fact_6031_of__int__le__numeral__iff,axiom,
! [Z: int,Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).
% of_int_le_numeral_iff
tff(fact_6032_of__int__le__numeral__iff,axiom,
! [Z: int,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).
% of_int_le_numeral_iff
tff(fact_6033_of__int__numeral__le__iff,axiom,
! [Nb: num,Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).
% of_int_numeral_le_iff
tff(fact_6034_of__int__numeral__le__iff,axiom,
! [Nb: num,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).
% of_int_numeral_le_iff
tff(fact_6035_of__int__numeral__le__iff,axiom,
! [Nb: num,Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).
% of_int_numeral_le_iff
tff(fact_6036_of__int__less__numeral__iff,axiom,
! [Z: int,Nb: num] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).
% of_int_less_numeral_iff
tff(fact_6037_of__int__less__numeral__iff,axiom,
! [Z: int,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).
% of_int_less_numeral_iff
tff(fact_6038_of__int__less__numeral__iff,axiom,
! [Z: int,Nb: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).
% of_int_less_numeral_iff
tff(fact_6039_of__int__numeral__less__iff,axiom,
! [Nb: num,Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).
% of_int_numeral_less_iff
tff(fact_6040_of__int__numeral__less__iff,axiom,
! [Nb: num,Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).
% of_int_numeral_less_iff
tff(fact_6041_of__int__numeral__less__iff,axiom,
! [Nb: num,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).
% of_int_numeral_less_iff
tff(fact_6042_of__int__le__1__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).
% of_int_le_1_iff
tff(fact_6043_of__int__le__1__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).
% of_int_le_1_iff
tff(fact_6044_of__int__le__1__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).
% of_int_le_1_iff
tff(fact_6045_of__int__1__le__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).
% of_int_1_le_iff
tff(fact_6046_of__int__1__le__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).
% of_int_1_le_iff
tff(fact_6047_of__int__1__le__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).
% of_int_1_le_iff
tff(fact_6048_of__int__less__1__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).
% of_int_less_1_iff
tff(fact_6049_of__int__less__1__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).
% of_int_less_1_iff
tff(fact_6050_of__int__less__1__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).
% of_int_less_1_iff
tff(fact_6051_of__int__1__less__iff,axiom,
! [Z: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% of_int_1_less_iff
tff(fact_6052_of__int__1__less__iff,axiom,
! [Z: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% of_int_1_less_iff
tff(fact_6053_of__int__1__less__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% of_int_1_less_iff
tff(fact_6054_numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_6055_numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_6056_numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_6057_numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_6058_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_6059_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_6060_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_6061_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_6062_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( aa_rat_o(aa_rat_fun_rat_o(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,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_6063_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( aa_int_o(aa_int_fun_int_o(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,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_6064_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( aa_real_o(aa_real_fun_real_o(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,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_6065_of__int__power__le__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_6066_of__int__power__le__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_6067_of__int__power__le__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_6068_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( aa_real_o(aa_real_fun_real_o(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,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_6069_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( aa_rat_o(aa_rat_fun_rat_o(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,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_6070_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,Xb: int] :
( aa_int_o(aa_int_fun_int_o(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,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_6071_of__int__power__less__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_6072_of__int__power__less__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_6073_of__int__power__less__of__int__cancel__iff,axiom,
! [Xb: int,B: int,W: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_6074_numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_6075_numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_6076_numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_6077_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_6078_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(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,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_6079_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(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_num_real(numeral_numeral_real,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_6080_numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_6081_numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_6082_numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_6083_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_6084_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_6085_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_6086_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = 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,Xb))),Nb) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6087_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) = 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,Xb))),Nb) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6088_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6089_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) = 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,Xb))),Nb) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6090_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) = 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,Xb))),Nb) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6091_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: 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,Xb))),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6092_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: 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,aa_num_real(numeral_numeral_real,Xb))),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6093_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6094_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6095_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6096_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6097_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6098_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6099_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6100_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6101_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6102_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(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,Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6103_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(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,aa_num_real(numeral_numeral_real,Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6104_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6105_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6106_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6107_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6108_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(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,Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6109_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(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,aa_num_real(numeral_numeral_real,Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6110_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6111_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6112_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_eh),Prod) ).
% prod.disc_eq_case
tff(fact_6113_Collect__case__prod__mono,axiom,
! [A3: fun_int_fun_int_o,B3: fun_int_fun_int_o] :
( ord_le77980682512674322_int_o(A3,B3)
=> ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A3)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B3))) ) ).
% Collect_case_prod_mono
tff(fact_6114_mult__of__int__commute,axiom,
! [Xb: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,Xb)) ).
% mult_of_int_commute
tff(fact_6115_mult__of__int__commute,axiom,
! [Xb: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,Xb)) ).
% mult_of_int_commute
tff(fact_6116_mult__of__int__commute,axiom,
! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,Xb)) ).
% mult_of_int_commute
tff(fact_6117_of__int__nonneg,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).
% of_int_nonneg
tff(fact_6118_of__int__nonneg,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).
% of_int_nonneg
tff(fact_6119_of__int__nonneg,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).
% of_int_nonneg
tff(fact_6120_of__int__pos,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).
% of_int_pos
tff(fact_6121_of__int__pos,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).
% of_int_pos
tff(fact_6122_of__int__pos,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).
% of_int_pos
tff(fact_6123_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_6124_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,aa_num_real(numeral_numeral_real,K)) ).
% of_int_neg_numeral
tff(fact_6125_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_6126_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))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).
% of_int_neg_numeral
tff(fact_6127_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_6128_int__le__real__less,axiom,
! [Nb: int,Mb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).
% int_le_real_less
tff(fact_6129_int__less__real__le,axiom,
! [Nb: int,Mb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).
% int_less_real_le
tff(fact_6130_even__of__int__iff,axiom,
! [K: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).
% even_of_int_iff
tff(fact_6131_even__of__int__iff,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).
% even_of_int_iff
tff(fact_6132_dbl__dec__def,axiom,
! [Xb: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xb) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).
% dbl_dec_def
tff(fact_6133_dbl__dec__def,axiom,
! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).
% dbl_dec_def
tff(fact_6134_dbl__dec__def,axiom,
! [Xb: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xb) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).
% dbl_dec_def
tff(fact_6135_dbl__dec__def,axiom,
! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).
% dbl_dec_def
tff(fact_6136_compl__le__swap2,axiom,
! [Y: set_nat,Xb: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),Xb)
=> aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),Y) ) ).
% compl_le_swap2
tff(fact_6137_compl__le__swap1,axiom,
! [Y: set_nat,Xb: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(Xb))
=> aa_set_nat_o(ord_less_eq_set_nat(Xb),uminus5710092332889474511et_nat(Y)) ) ).
% compl_le_swap1
tff(fact_6138_compl__mono,axiom,
! [Xb: set_nat,Y: set_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
=> aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(Xb)) ) ).
% compl_mono
tff(fact_6139_diff__shunt__var,axiom,
! [Xb: set_int,Y: set_int] :
( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
<=> aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).
% diff_shunt_var
tff(fact_6140_diff__shunt__var,axiom,
! [Xb: set_real,Y: set_real] :
( ( minus_minus_set_real(Xb,Y) = bot_bot_set_real )
<=> aa_set_real_o(ord_less_eq_set_real(Xb),Y) ) ).
% diff_shunt_var
tff(fact_6141_diff__shunt__var,axiom,
! [Xb: set_nat,Y: set_nat] :
( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
<=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).
% diff_shunt_var
tff(fact_6142_floor__exists,axiom,
! [Xb: rat] :
? [Z2: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).
% floor_exists
tff(fact_6143_floor__exists,axiom,
! [Xb: real] :
? [Z2: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).
% floor_exists
tff(fact_6144_floor__exists1,axiom,
! [Xb: rat] :
? [X4: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int)))
& ! [Y4: int] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
=> ( Y4 = X4 ) ) ) ).
% floor_exists1
tff(fact_6145_floor__exists1,axiom,
! [Xb: real] :
? [X4: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int)))
& ! [Y4: int] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
=> ( Y4 = X4 ) ) ) ).
% floor_exists1
tff(fact_6146_ln__one__minus__pos__lower__bound,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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),Xb))) ) ) ).
% ln_one_minus_pos_lower_bound
tff(fact_6147_divmod__BitM__2__eq,axiom,
! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),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,Mb)),one_one_int)),one_one_int) ).
% divmod_BitM_2_eq
tff(fact_6148_vebt__buildup_Opelims,axiom,
! [Xb: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_buildup(Xb) = Y )
=> ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
=> ( ( ( Xb = zero_zero_nat )
=> ( ( Y = vEBT_Leaf($false,$false) )
=> ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
=> ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
=> ( ( Y = vEBT_Leaf($false,$false) )
=> ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
=> ~ ! [Va2: nat] :
( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
=> ( ( Y = $ite(
aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
$let(
half: nat,
half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
$let(
half: nat,
half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),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,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
=> ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).
% vebt_buildup.pelims
tff(fact_6149_int__ge__less__than2__def,axiom,
! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ei(D))) ).
% int_ge_less_than2_def
tff(fact_6150_ln__one,axiom,
aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).
% ln_one
tff(fact_6151_ln__less__cancel__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).
% ln_less_cancel_iff
tff(fact_6152_ln__inj__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(ln_ln_real,Y) )
<=> ( Xb = Y ) ) ) ) ).
% ln_inj_iff
tff(fact_6153_dbl__dec__simps_I5_J,axiom,
! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bitM,K)) ).
% dbl_dec_simps(5)
tff(fact_6154_dbl__dec__simps_I5_J,axiom,
! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K)) ).
% dbl_dec_simps(5)
tff(fact_6155_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,aa_num_num(bitM,K)) ).
% dbl_dec_simps(5)
tff(fact_6156_ln__le__cancel__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).
% ln_le_cancel_iff
tff(fact_6157_ln__less__zero__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).
% ln_less_zero_iff
tff(fact_6158_ln__gt__zero__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).
% ln_gt_zero_iff
tff(fact_6159_ln__eq__zero__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( ( aa_real_real(ln_ln_real,Xb) = zero_zero_real )
<=> ( Xb = one_one_real ) ) ) ).
% ln_eq_zero_iff
tff(fact_6160_pred__numeral__simps_I2_J,axiom,
! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).
% pred_numeral_simps(2)
tff(fact_6161_ln__ge__zero__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).
% ln_ge_zero_iff
tff(fact_6162_ln__le__zero__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).
% ln_le_zero_iff
tff(fact_6163_semiring__norm_I26_J,axiom,
aa_num_num(bitM,one) = one ).
% semiring_norm(26)
tff(fact_6164_ln__less__self,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).
% ln_less_self
tff(fact_6165_semiring__norm_I27_J,axiom,
! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).
% semiring_norm(27)
tff(fact_6166_semiring__norm_I28_J,axiom,
! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).
% semiring_norm(28)
tff(fact_6167_ln__bound,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).
% ln_bound
tff(fact_6168_ln__gt__zero__imp__gt__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).
% ln_gt_zero_imp_gt_one
tff(fact_6169_ln__less__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real) ) ) ).
% ln_less_zero
tff(fact_6170_ln__gt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).
% ln_gt_zero
tff(fact_6171_eval__nat__numeral_I2_J,axiom,
! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).
% eval_nat_numeral(2)
tff(fact_6172_one__plus__BitM,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).
% one_plus_BitM
tff(fact_6173_BitM__plus__one,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).
% BitM_plus_one
tff(fact_6174_ln__ge__zero__imp__ge__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).
% ln_ge_zero_imp_ge_one
tff(fact_6175_ln__mult,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).
% ln_mult
tff(fact_6176_ln__eq__minus__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real) )
=> ( Xb = one_one_real ) ) ) ).
% ln_eq_minus_one
tff(fact_6177_ln__div,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).
% ln_div
tff(fact_6178_ln__2__less__1,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).
% ln_2_less_1
tff(fact_6179_numeral__BitM,axiom,
! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,Nb)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb))),one_one_rat) ).
% numeral_BitM
tff(fact_6180_numeral__BitM,axiom,
! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bitM,Nb)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).
% numeral_BitM
tff(fact_6181_numeral__BitM,axiom,
! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb))),one_one_real) ).
% numeral_BitM
tff(fact_6182_numeral__BitM,axiom,
! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).
% numeral_BitM
tff(fact_6183_odd__numeral__BitM,axiom,
! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bitM,W))) ).
% odd_numeral_BitM
tff(fact_6184_odd__numeral__BitM,axiom,
! [W: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,W))) ).
% odd_numeral_BitM
tff(fact_6185_odd__numeral__BitM,axiom,
! [W: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bitM,W))) ).
% odd_numeral_BitM
tff(fact_6186_ln__le__minus__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ).
% ln_le_minus_one
tff(fact_6187_ln__diff__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),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,Xb),Y)),Y)) ) ) ).
% ln_diff_le
tff(fact_6188_ln__add__one__self__le__self2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).
% ln_add_one_self_le_self2
tff(fact_6189_ln__one__minus__pos__upper__bound,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).
% ln_one_minus_pos_upper_bound
tff(fact_6190_exists__least__lemma,axiom,
! [P: fun_nat_o] :
( ~ aa_nat_o(P,zero_zero_nat)
=> ( ? [X_1: nat] : aa_nat_o(P,X_1)
=> ? [N: nat] :
( ~ aa_nat_o(P,N)
& aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).
% exists_least_lemma
tff(fact_6191_ex__le__of__int,axiom,
! [Xb: rat] :
? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).
% ex_le_of_int
tff(fact_6192_ex__le__of__int,axiom,
! [Xb: real] :
? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).
% ex_le_of_int
tff(fact_6193_ex__of__int__less,axiom,
! [Xb: real] :
? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xb) ).
% ex_of_int_less
tff(fact_6194_ex__of__int__less,axiom,
! [Xb: rat] :
? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb) ).
% ex_of_int_less
tff(fact_6195_ex__less__of__int,axiom,
! [Xb: real] :
? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).
% ex_less_of_int
tff(fact_6196_ex__less__of__int,axiom,
! [Xb: rat] :
? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).
% ex_less_of_int
tff(fact_6197_ln__one__plus__pos__lower__bound,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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),Xb))) ) ) ).
% ln_one_plus_pos_lower_bound
tff(fact_6198_artanh__def,axiom,
! [Xb: real] : aa_real_real(artanh_real,Xb) = 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),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% artanh_def
tff(fact_6199_int__ge__less__than__def,axiom,
! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ej(D))) ).
% int_ge_less_than_def
tff(fact_6200_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6201_round__unique,axiom,
! [Xb: rat,Y: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
=> ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Y ) ) ) ).
% round_unique
tff(fact_6202_round__unique,axiom,
! [Xb: real,Y: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
=> ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).
% round_unique
tff(fact_6203_tanh__ln__real,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = 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,Xb),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).
% tanh_ln_real
tff(fact_6204_pred__subset__eq2,axiom,
! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
( ord_le2909477399203599485un_o_o(aTP_Lamm_ek(R),aTP_Lamm_ek(S3))
<=> ord_le8980329558974975238eger_o(R,S3) ) ).
% pred_subset_eq2
tff(fact_6205_pred__subset__eq2,axiom,
! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
( ord_le3306795289226699318_num_o(aTP_Lamm_el(R),aTP_Lamm_el(S3))
<=> ord_le880128212290418581um_num(R,S3) ) ).
% pred_subset_eq2
tff(fact_6206_pred__subset__eq2,axiom,
! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
( ord_le1526520017675560704_num_o(aTP_Lamm_em(R),aTP_Lamm_em(S3))
<=> ord_le8085105155179020875at_num(R,S3) ) ).
% pred_subset_eq2
tff(fact_6207_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ord_le768339454706049738_nat_o(aTP_Lamm_en(R),aTP_Lamm_en(S3))
<=> ord_le3146513528884898305at_nat(R,S3) ) ).
% pred_subset_eq2
tff(fact_6208_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ord_le77980682512674322_int_o(aTP_Lamm_eo(R),aTP_Lamm_eo(S3))
<=> ord_le2843351958646193337nt_int(R,S3) ) ).
% pred_subset_eq2
tff(fact_6209_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_6210_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_6211_abs__idempotent,axiom,
! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).
% abs_idempotent
tff(fact_6212_abs__idempotent,axiom,
! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).
% abs_idempotent
tff(fact_6213_abs__zero,axiom,
abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% abs_zero
tff(fact_6214_abs__zero,axiom,
aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).
% abs_zero
tff(fact_6215_abs__zero,axiom,
aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).
% abs_zero
tff(fact_6216_abs__zero,axiom,
aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
% abs_zero
tff(fact_6217_abs__eq__0,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
tff(fact_6218_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_6219_abs__eq__0,axiom,
! [A: rat] :
( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% abs_eq_0
tff(fact_6220_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_6221_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
tff(fact_6222_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_6223_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
<=> ( A = zero_zero_rat ) ) ).
% abs_0_eq
tff(fact_6224_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_6225_abs__numeral,axiom,
! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).
% abs_numeral
tff(fact_6226_abs__numeral,axiom,
! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).
% abs_numeral
tff(fact_6227_abs__numeral,axiom,
! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).
% abs_numeral
tff(fact_6228_abs__numeral,axiom,
! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).
% abs_numeral
tff(fact_6229_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_6230_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_6231_abs__mult__self__eq,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).
% abs_mult_self_eq
tff(fact_6232_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_6233_abs__1,axiom,
abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).
% abs_1
tff(fact_6234_abs__1,axiom,
aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).
% abs_1
tff(fact_6235_abs__1,axiom,
aa_real_real(abs_abs_real,one_one_real) = one_one_real ).
% abs_1
tff(fact_6236_abs__1,axiom,
aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).
% abs_1
tff(fact_6237_abs__1,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_1
tff(fact_6238_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_6239_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_6240_abs__add__abs,axiom,
! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).
% abs_add_abs
tff(fact_6241_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_6242_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_6243_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_6244_abs__minus__cancel,axiom,
! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).
% abs_minus_cancel
tff(fact_6245_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_6246_tanh__real__less__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% tanh_real_less_iff
tff(fact_6247_abs__of__nonneg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( abs_abs_Code_integer(A) = A ) ) ).
% abs_of_nonneg
tff(fact_6248_abs__of__nonneg,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).
% abs_of_nonneg
tff(fact_6249_abs__of__nonneg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_nonneg
tff(fact_6250_abs__of__nonneg,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_real(abs_abs_real,A) = A ) ) ).
% abs_of_nonneg
tff(fact_6251_abs__le__self__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% abs_le_self_iff
tff(fact_6252_abs__le__self__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% abs_le_self_iff
tff(fact_6253_abs__le__self__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% abs_le_self_iff
tff(fact_6254_abs__le__self__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% abs_le_self_iff
tff(fact_6255_abs__le__zero__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
tff(fact_6256_abs__le__zero__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
<=> ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
tff(fact_6257_abs__le__zero__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
<=> ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
tff(fact_6258_abs__le__zero__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
<=> ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
tff(fact_6259_zero__less__abs__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
<=> ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
tff(fact_6260_zero__less__abs__iff,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
<=> ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
tff(fact_6261_zero__less__abs__iff,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
<=> ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
tff(fact_6262_zero__less__abs__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
<=> ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
tff(fact_6263_abs__neg__numeral,axiom,
! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).
% abs_neg_numeral
tff(fact_6264_abs__neg__numeral,axiom,
! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).
% abs_neg_numeral
tff(fact_6265_abs__neg__numeral,axiom,
! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).
% abs_neg_numeral
tff(fact_6266_abs__neg__numeral,axiom,
! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).
% abs_neg_numeral
tff(fact_6267_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_6268_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_6269_abs__neg__one,axiom,
aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).
% abs_neg_one
tff(fact_6270_abs__neg__one,axiom,
abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).
% abs_neg_one
tff(fact_6271_abs__power__minus,axiom,
! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) ).
% abs_power_minus
tff(fact_6272_abs__power__minus,axiom,
! [A: real,Nb: 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)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% abs_power_minus
tff(fact_6273_abs__power__minus,axiom,
! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) ).
% abs_power_minus
tff(fact_6274_abs__power__minus,axiom,
! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).
% abs_power_minus
tff(fact_6275_tanh__real__pos__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% tanh_real_pos_iff
tff(fact_6276_tanh__real__neg__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).
% tanh_real_neg_iff
tff(fact_6277_round__numeral,axiom,
! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).
% round_numeral
tff(fact_6278_round__1,axiom,
aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).
% round_1
tff(fact_6279_round__1,axiom,
aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).
% round_1
tff(fact_6280_divide__le__0__abs__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
| ( B = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
tff(fact_6281_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
tff(fact_6282_zero__le__divide__abs__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B)))
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
| ( B = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
tff(fact_6283_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
tff(fact_6284_abs__of__nonpos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_nonpos
tff(fact_6285_abs__of__nonpos,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
=> ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% abs_of_nonpos
tff(fact_6286_abs__of__nonpos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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_6287_abs__of__nonpos,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(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_6288_artanh__minus__real,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xb)) ) ) ).
% artanh_minus_real
tff(fact_6289_zero__less__power__abs__iff,axiom,
! [A: code_integer,Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
<=> ( ( A != zero_z3403309356797280102nteger )
| ( Nb = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_6290_zero__less__power__abs__iff,axiom,
! [A: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
<=> ( ( A != zero_zero_real )
| ( Nb = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_6291_zero__less__power__abs__iff,axiom,
! [A: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
<=> ( ( A != zero_zero_rat )
| ( Nb = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_6292_zero__less__power__abs__iff,axiom,
! [A: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb))
<=> ( ( A != zero_zero_int )
| ( Nb = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_6293_abs__power2,axiom,
! [A: rat] : aa_rat_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_6294_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_6295_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_6296_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_6297_power2__abs,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_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_6298_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_6299_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_6300_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_6301_round__neg__numeral,axiom,
! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).
% round_neg_numeral
tff(fact_6302_round__neg__numeral,axiom,
! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).
% round_neg_numeral
tff(fact_6303_power__even__abs__numeral,axiom,
! [W: num,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(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(aa_rat_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_6304_power__even__abs__numeral,axiom,
! [W: num,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(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_6305_power__even__abs__numeral,axiom,
! [W: num,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(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_6306_power__even__abs__numeral,axiom,
! [W: num,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(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_6307_abs__ge__self,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).
% abs_ge_self
tff(fact_6308_abs__ge__self,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).
% abs_ge_self
tff(fact_6309_abs__ge__self,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).
% abs_ge_self
tff(fact_6310_abs__ge__self,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).
% abs_ge_self
tff(fact_6311_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% abs_le_D1
tff(fact_6312_abs__le__D1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).
% abs_le_D1
tff(fact_6313_abs__le__D1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% abs_le_D1
tff(fact_6314_abs__le__D1,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).
% abs_le_D1
tff(fact_6315_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_6316_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_6317_abs__mult,axiom,
! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).
% abs_mult
tff(fact_6318_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_6319_abs__one,axiom,
abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).
% abs_one
tff(fact_6320_abs__one,axiom,
aa_real_real(abs_abs_real,one_one_real) = one_one_real ).
% abs_one
tff(fact_6321_abs__one,axiom,
aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).
% abs_one
tff(fact_6322_abs__one,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_one
tff(fact_6323_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_6324_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_6325_abs__minus__commute,axiom,
! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).
% abs_minus_commute
tff(fact_6326_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_6327_power__abs,axiom,
! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) ).
% power_abs
tff(fact_6328_power__abs,axiom,
! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).
% power_abs
tff(fact_6329_power__abs,axiom,
! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) ).
% power_abs
tff(fact_6330_power__abs,axiom,
! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) ).
% power_abs
tff(fact_6331_round__diff__minimal,axiom,
! [Z: rat,Mb: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,Mb)))) ).
% round_diff_minimal
tff(fact_6332_round__diff__minimal,axiom,
! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(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,Mb)))) ).
% round_diff_minimal
tff(fact_6333_abs__ge__zero,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).
% abs_ge_zero
tff(fact_6334_abs__ge__zero,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).
% abs_ge_zero
tff(fact_6335_abs__ge__zero,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).
% abs_ge_zero
tff(fact_6336_abs__ge__zero,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).
% abs_ge_zero
tff(fact_6337_abs__not__less__zero,axiom,
! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).
% abs_not_less_zero
tff(fact_6338_abs__not__less__zero,axiom,
! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).
% abs_not_less_zero
tff(fact_6339_abs__not__less__zero,axiom,
! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).
% abs_not_less_zero
tff(fact_6340_abs__not__less__zero,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).
% abs_not_less_zero
tff(fact_6341_abs__of__pos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( abs_abs_Code_integer(A) = A ) ) ).
% abs_of_pos
tff(fact_6342_abs__of__pos,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_real(abs_abs_real,A) = A ) ) ).
% abs_of_pos
tff(fact_6343_abs__of__pos,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).
% abs_of_pos
tff(fact_6344_abs__of__pos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_pos
tff(fact_6345_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6346_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).
% abs_triangle_ineq
tff(fact_6347_abs__triangle__ineq,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_6348_abs__triangle__ineq,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,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_6349_abs__mult__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6350_abs__mult__less,axiom,
! [A: real,C: real,B: real,D: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
=> aa_real_o(aa_real_fun_real_o(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_6351_abs__mult__less,axiom,
! [A: rat,C: rat,B: rat,D: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).
% abs_mult_less
tff(fact_6352_abs__mult__less,axiom,
! [A: int,C: int,B: int,D: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
=> aa_int_o(aa_int_fun_int_o(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_6353_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6354_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).
% abs_triangle_ineq2
tff(fact_6355_abs__triangle__ineq2,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_6356_abs__triangle__ineq2,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_6357_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6358_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).
% abs_triangle_ineq3
tff(fact_6359_abs__triangle__ineq3,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_6360_abs__triangle__ineq3,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(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_6361_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6362_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).
% abs_triangle_ineq2_sym
tff(fact_6363_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_6364_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_6365_abs__ge__minus__self,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).
% abs_ge_minus_self
tff(fact_6366_abs__ge__minus__self,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).
% abs_ge_minus_self
tff(fact_6367_abs__ge__minus__self,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).
% abs_ge_minus_self
tff(fact_6368_abs__ge__minus__self,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).
% abs_ge_minus_self
tff(fact_6369_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).
% abs_le_iff
tff(fact_6370_abs__le__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).
% abs_le_iff
tff(fact_6371_abs__le__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).
% abs_le_iff
tff(fact_6372_abs__le__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).
% abs_le_iff
tff(fact_6373_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).
% abs_le_D2
tff(fact_6374_abs__le__D2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).
% abs_le_D2
tff(fact_6375_abs__le__D2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).
% abs_le_D2
tff(fact_6376_abs__le__D2,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).
% abs_le_D2
tff(fact_6377_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).
% abs_leI
tff(fact_6378_abs__leI,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).
% abs_leI
tff(fact_6379_abs__leI,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).
% abs_leI
tff(fact_6380_abs__leI,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).
% abs_leI
tff(fact_6381_abs__less__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).
% abs_less_iff
tff(fact_6382_abs__less__iff,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).
% abs_less_iff
tff(fact_6383_abs__less__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).
% abs_less_iff
tff(fact_6384_abs__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).
% abs_less_iff
tff(fact_6385_tanh__real__lt__1,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),one_one_real) ).
% tanh_real_lt_1
tff(fact_6386_dense__eq0__I,axiom,
! [Xb: rat] :
( ! [E2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),E2) )
=> ( Xb = zero_zero_rat ) ) ).
% dense_eq0_I
tff(fact_6387_dense__eq0__I,axiom,
! [Xb: real] :
( ! [E2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),E2) )
=> ( Xb = zero_zero_real ) ) ).
% dense_eq0_I
tff(fact_6388_abs__mult__pos,axiom,
! [Xb: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb)) ) ) ).
% abs_mult_pos
tff(fact_6389_abs__mult__pos,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
=> ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),Xb)) ) ) ).
% abs_mult_pos
tff(fact_6390_abs__mult__pos,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)) ) ) ).
% abs_mult_pos
tff(fact_6391_abs__mult__pos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),Xb) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)) ) ) ).
% abs_mult_pos
tff(fact_6392_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6393_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
& ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
=> ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).
% abs_eq_mult
tff(fact_6394_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
| aa_int_o(aa_int_fun_int_o(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_6395_abs__eq__mult,axiom,
! [A: real,B: real] :
( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
| aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
| aa_real_o(aa_real_fun_real_o(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_6396_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer(A) = B )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
& ( ( A = B )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6397_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(abs_abs_rat,A) = B )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
& ( ( A = B )
| ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6398_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( aa_int_int(abs_abs_int,A) = B )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
& ( ( A = B )
| ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6399_abs__eq__iff_H,axiom,
! [A: real,B: real] :
( ( aa_real_real(abs_abs_real,A) = B )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
& ( ( A = B )
| ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6400_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A = abs_abs_Code_integer(B) )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
& ( ( B = A )
| ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6401_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(abs_abs_rat,B) )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
& ( ( B = A )
| ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6402_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(abs_abs_int,B) )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& ( ( B = A )
| ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6403_eq__abs__iff_H,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(abs_abs_real,B) )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
& ( ( B = A )
| ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6404_abs__minus__le__zero,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).
% abs_minus_le_zero
tff(fact_6405_abs__minus__le__zero,axiom,
! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).
% abs_minus_le_zero
tff(fact_6406_abs__minus__le__zero,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(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_6407_abs__minus__le__zero,axiom,
! [A: real] : aa_real_o(aa_real_fun_real_o(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_6408_abs__div__pos,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xb)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).
% abs_div_pos
tff(fact_6409_abs__div__pos,axiom,
! [Y: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).
% abs_div_pos
tff(fact_6410_zero__le__power__abs,axiom,
! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).
% zero_le_power_abs
tff(fact_6411_zero__le__power__abs,axiom,
! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).
% zero_le_power_abs
tff(fact_6412_zero__le__power__abs,axiom,
! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb)) ).
% zero_le_power_abs
tff(fact_6413_zero__le__power__abs,axiom,
! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(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)),Nb)) ).
% zero_le_power_abs
tff(fact_6414_abs__if,axiom,
! [A: int] :
aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).
% abs_if
tff(fact_6415_abs__if,axiom,
! [A: real] :
aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).
% abs_if
tff(fact_6416_abs__if,axiom,
! [A: rat] :
aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).
% abs_if
tff(fact_6417_abs__if,axiom,
! [A: code_integer] :
abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).
% abs_if
tff(fact_6418_abs__if__raw,axiom,
! [X3: int] :
aa_int_int(abs_abs_int,X3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int),aa_int_int(uminus_uminus_int,X3),X3) ).
% abs_if_raw
tff(fact_6419_abs__if__raw,axiom,
! [X3: real] :
aa_real_real(abs_abs_real,X3) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real),aa_real_real(uminus_uminus_real,X3),X3) ).
% abs_if_raw
tff(fact_6420_abs__if__raw,axiom,
! [X3: rat] :
aa_rat_rat(abs_abs_rat,X3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X3),X3) ).
% abs_if_raw
tff(fact_6421_abs__if__raw,axiom,
! [X3: code_integer] :
abs_abs_Code_integer(X3) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3),X3) ).
% abs_if_raw
tff(fact_6422_abs__of__neg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(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_6423_abs__of__neg,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(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_6424_abs__of__neg,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% abs_of_neg
tff(fact_6425_abs__of__neg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_neg
tff(fact_6426_abs__diff__le__iff,axiom,
! [Xb: code_integer,A: code_integer,R2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xb)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).
% abs_diff_le_iff
tff(fact_6427_abs__diff__le__iff,axiom,
! [Xb: rat,A: rat,R2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).
% abs_diff_le_iff
tff(fact_6428_abs__diff__le__iff,axiom,
! [Xb: int,A: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).
% abs_diff_le_iff
tff(fact_6429_abs__diff__le__iff,axiom,
! [Xb: real,A: real,R2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).
% abs_diff_le_iff
tff(fact_6430_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6431_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).
% abs_diff_triangle_ineq
tff(fact_6432_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(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_6433_abs__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(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_6434_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_6435_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).
% abs_triangle_ineq4
tff(fact_6436_abs__triangle__ineq4,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_6437_abs__triangle__ineq4,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(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_6438_abs__diff__less__iff,axiom,
! [Xb: code_integer,A: code_integer,R2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xb)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).
% abs_diff_less_iff
tff(fact_6439_abs__diff__less__iff,axiom,
! [Xb: real,A: real,R2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).
% abs_diff_less_iff
tff(fact_6440_abs__diff__less__iff,axiom,
! [Xb: rat,A: rat,R2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).
% abs_diff_less_iff
tff(fact_6441_abs__diff__less__iff,axiom,
! [Xb: int,A: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).
% abs_diff_less_iff
tff(fact_6442_round__mono,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xb)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).
% round_mono
tff(fact_6443_round__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xb)),aa_real_int(archim8280529875227126926d_real,Y)) ) ).
% round_mono
tff(fact_6444_abs__real__def,axiom,
! [A: real] :
aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).
% abs_real_def
tff(fact_6445_lemma__interval__lt,axiom,
! [A: real,Xb: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [Y4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B) ) ) ) ) ) ).
% lemma_interval_lt
tff(fact_6446_tanh__real__gt__neg1,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xb)) ).
% tanh_real_gt_neg1
tff(fact_6447_abs__add__one__gt__zero,axiom,
! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).
% abs_add_one_gt_zero
tff(fact_6448_abs__add__one__gt__zero,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xb))) ).
% abs_add_one_gt_zero
tff(fact_6449_abs__add__one__gt__zero,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xb))) ).
% abs_add_one_gt_zero
tff(fact_6450_abs__add__one__gt__zero,axiom,
! [Xb: int] : aa_int_o(aa_int_fun_int_o(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,Xb))) ).
% abs_add_one_gt_zero
tff(fact_6451_of__int__leD,axiom,
! [Nb: int,Xb: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).
% of_int_leD
tff(fact_6452_of__int__leD,axiom,
! [Nb: int,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).
% of_int_leD
tff(fact_6453_of__int__leD,axiom,
! [Nb: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).
% of_int_leD
tff(fact_6454_of__int__leD,axiom,
! [Nb: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).
% of_int_leD
tff(fact_6455_of__int__lessD,axiom,
! [Nb: int,Xb: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).
% of_int_lessD
tff(fact_6456_of__int__lessD,axiom,
! [Nb: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).
% of_int_lessD
tff(fact_6457_of__int__lessD,axiom,
! [Nb: int,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).
% of_int_lessD
tff(fact_6458_of__int__lessD,axiom,
! [Nb: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
=> ( ( Nb = zero_zero_int )
| aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).
% of_int_lessD
tff(fact_6459_lemma__interval,axiom,
! [A: real,Xb: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [Y4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B) ) ) ) ) ) ).
% lemma_interval
tff(fact_6460_of__int__round__abs__le,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),Xb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).
% of_int_round_abs_le
tff(fact_6461_of__int__round__abs__le,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% of_int_round_abs_le
tff(fact_6462_round__unique_H,axiom,
! [Xb: real,Nb: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_int(archim8280529875227126926d_real,Xb) = Nb ) ) ).
% round_unique'
tff(fact_6463_round__unique_H,axiom,
! [Xb: rat,Nb: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Nb)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
=> ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Nb ) ) ).
% round_unique'
tff(fact_6464_abs__le__square__iff,axiom,
! [Xb: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),abs_abs_Code_integer(Y))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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_6465_abs__le__square__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),aa_rat_rat(abs_abs_rat,Y))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),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_6466_abs__le__square__iff,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),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_6467_abs__le__square__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_6468_pred__equals__eq2,axiom,
! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
( ! [X: code_integer,Xa2: $o] :
( member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa2)),R)
<=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa2)),S3) )
<=> ( R = S3 ) ) ).
% pred_equals_eq2
tff(fact_6469_pred__equals__eq2,axiom,
! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
( ! [X: num,Xa2: num] :
( member7279096912039735102um_num(product_Pair_num_num(X,Xa2),R)
<=> member7279096912039735102um_num(product_Pair_num_num(X,Xa2),S3) )
<=> ( R = S3 ) ) ).
% pred_equals_eq2
tff(fact_6470_pred__equals__eq2,axiom,
! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
( ! [X: nat,Xa2: num] :
( member9148766508732265716at_num(product_Pair_nat_num(X,Xa2),R)
<=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa2),S3) )
<=> ( R = S3 ) ) ).
% pred_equals_eq2
tff(fact_6471_pred__equals__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Xa2: nat] :
( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2),R)
<=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2),S3) )
<=> ( R = S3 ) ) ).
% pred_equals_eq2
tff(fact_6472_pred__equals__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ! [X: int,Xa2: int] :
( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2),R)
<=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2),S3) )
<=> ( R = S3 ) ) ).
% pred_equals_eq2
tff(fact_6473_bot__empty__eq2,axiom,
! [X3: code_integer,Xa: $o] :
( aa_o_o(aa_Cod6379977994291289310un_o_o(bot_bo897209107778963633un_o_o,X3),(Xa))
<=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa)),bot_bo5379713665208646970eger_o) ) ).
% bot_empty_eq2
tff(fact_6474_bot__empty__eq2,axiom,
! [X3: num,Xa: num] :
( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X3),Xa)
<=> member7279096912039735102um_num(product_Pair_num_num(X3,Xa),bot_bo9056780473022590049um_num) ) ).
% bot_empty_eq2
tff(fact_6475_bot__empty__eq2,axiom,
! [X3: nat,Xa: num] :
( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X3),Xa)
<=> member9148766508732265716at_num(product_Pair_nat_num(X3,Xa),bot_bo7038385379056416535at_num) ) ).
% bot_empty_eq2
tff(fact_6476_bot__empty__eq2,axiom,
! [X3: nat,Xa: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X3),Xa)
<=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa),bot_bo2099793752762293965at_nat) ) ).
% bot_empty_eq2
tff(fact_6477_bot__empty__eq2,axiom,
! [X3: int,Xa: int] :
( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X3),Xa)
<=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa),bot_bo1796632182523588997nt_int) ) ).
% bot_empty_eq2
tff(fact_6478_abs__square__eq__1,axiom,
! [Xb: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
<=> ( abs_abs_Code_integer(Xb) = one_one_Code_integer ) ) ).
% abs_square_eq_1
tff(fact_6479_abs__square__eq__1,axiom,
! [Xb: rat] :
( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
<=> ( aa_rat_rat(abs_abs_rat,Xb) = one_one_rat ) ) ).
% abs_square_eq_1
tff(fact_6480_abs__square__eq__1,axiom,
! [Xb: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
<=> ( aa_real_real(abs_abs_real,Xb) = one_one_real ) ) ).
% abs_square_eq_1
tff(fact_6481_abs__square__eq__1,axiom,
! [Xb: int] :
( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
<=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).
% abs_square_eq_1
tff(fact_6482_power__even__abs,axiom,
! [Nb: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).
% power_even_abs
tff(fact_6483_power__even__abs,axiom,
! [Nb: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).
% power_even_abs
tff(fact_6484_power__even__abs,axiom,
! [Nb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).
% power_even_abs
tff(fact_6485_power__even__abs,axiom,
! [Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).
% power_even_abs
tff(fact_6486_abs__sqrt__wlog,axiom,
! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
( ! [X4: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).
% abs_sqrt_wlog
tff(fact_6487_abs__sqrt__wlog,axiom,
! [P: fun_rat_fun_rat_o,Xb: rat] :
( ! [X4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X4)
=> aa_rat_o(aa_rat_fun_rat_o(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
=> aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).
% abs_sqrt_wlog
tff(fact_6488_abs__sqrt__wlog,axiom,
! [P: fun_int_fun_int_o,Xb: int] :
( ! [X4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
=> aa_int_o(aa_int_fun_int_o(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
=> aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xb)),aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).
% abs_sqrt_wlog
tff(fact_6489_abs__sqrt__wlog,axiom,
! [P: fun_real_fun_real_o,Xb: real] :
( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
=> aa_real_o(aa_real_fun_real_o(P,X4),aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
=> aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).
% abs_sqrt_wlog
tff(fact_6490_power2__le__iff__abs__le,axiom,
! [Y: code_integer,Xb: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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))))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),Y) ) ) ).
% power2_le_iff_abs_le
tff(fact_6491_power2__le__iff__abs__le,axiom,
! [Y: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),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_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Y) ) ) ).
% power2_le_iff_abs_le
tff(fact_6492_power2__le__iff__abs__le,axiom,
! [Y: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),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_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y) ) ) ).
% power2_le_iff_abs_le
tff(fact_6493_power2__le__iff__abs__le,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y) ) ) ).
% power2_le_iff_abs_le
tff(fact_6494_abs__square__le__1,axiom,
! [Xb: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).
% abs_square_le_1
tff(fact_6495_abs__square__le__1,axiom,
! [Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).
% abs_square_le_1
tff(fact_6496_abs__square__le__1,axiom,
! [Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).
% abs_square_le_1
tff(fact_6497_abs__square__le__1,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).
% abs_square_le_1
tff(fact_6498_abs__square__less__1,axiom,
! [Xb: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).
% abs_square_less_1
tff(fact_6499_abs__square__less__1,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).
% abs_square_less_1
tff(fact_6500_abs__square__less__1,axiom,
! [Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).
% abs_square_less_1
tff(fact_6501_abs__square__less__1,axiom,
! [Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).
% abs_square_less_1
tff(fact_6502_power__mono__even,axiom,
! [Nb: nat,A: code_integer,B: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).
% power_mono_even
tff(fact_6503_power__mono__even,axiom,
! [Nb: nat,A: rat,B: rat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).
% power_mono_even
tff(fact_6504_power__mono__even,axiom,
! [Nb: nat,A: int,B: int] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).
% power_mono_even
tff(fact_6505_power__mono__even,axiom,
! [Nb: nat,A: real,B: real] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).
% power_mono_even
tff(fact_6506_subrelI,axiom,
! [R2: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
( ! [X4: code_integer,Y3: $o] :
( member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Y3)),R2)
=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Y3)),S2) )
=> ord_le8980329558974975238eger_o(R2,S2) ) ).
% subrelI
tff(fact_6507_subrelI,axiom,
! [R2: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
( ! [X4: num,Y3: num] :
( member7279096912039735102um_num(product_Pair_num_num(X4,Y3),R2)
=> member7279096912039735102um_num(product_Pair_num_num(X4,Y3),S2) )
=> ord_le880128212290418581um_num(R2,S2) ) ).
% subrelI
tff(fact_6508_subrelI,axiom,
! [R2: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
( ! [X4: nat,Y3: num] :
( member9148766508732265716at_num(product_Pair_nat_num(X4,Y3),R2)
=> member9148766508732265716at_num(product_Pair_nat_num(X4,Y3),S2) )
=> ord_le8085105155179020875at_num(R2,S2) ) ).
% subrelI
tff(fact_6509_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X4: nat,Y3: nat] :
( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3),R2)
=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3),S2) )
=> ord_le3146513528884898305at_nat(R2,S2) ) ).
% subrelI
tff(fact_6510_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ! [X4: int,Y3: int] :
( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3),R2)
=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3),S2) )
=> ord_le2843351958646193337nt_int(R2,S2) ) ).
% subrelI
tff(fact_6511_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6512_of__int__round__le,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).
% of_int_round_le
tff(fact_6513_of__int__round__le,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% of_int_round_le
tff(fact_6514_of__int__round__ge,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).
% of_int_round_ge
tff(fact_6515_of__int__round__ge,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).
% of_int_round_ge
tff(fact_6516_of__int__round__gt,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).
% of_int_round_gt
tff(fact_6517_of__int__round__gt,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).
% of_int_round_gt
tff(fact_6518_abs__ln__one__plus__x__minus__x__bound,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(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),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_ln_one_plus_x_minus_x_bound
tff(fact_6519_arctan__double,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xb)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% arctan_double
tff(fact_6520_add__scale__eq__noteq,axiom,
! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6521_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C: real,D: real] :
( ( R2 != 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,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6522_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6523_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != 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,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6524_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != 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,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6525_Sum__Icc__int,axiom,
! [Mb: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
=> ( groups4538972089207619220nt_int(aTP_Lamm_ak,set_or1266510415728281911st_int(Mb,Nb)) = aa_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,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% Sum_Icc_int
tff(fact_6526_even__set__encode__iff,axiom,
! [A3: set_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A3))
<=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A3) ) ) ).
% even_set_encode_iff
tff(fact_6527_mask__numeral,axiom,
! [Nb: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Nb)) = 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,Nb)))) ).
% mask_numeral
tff(fact_6528_mask__numeral,axiom,
! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = 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,Nb)))) ).
% mask_numeral
tff(fact_6529_num_Osize__gen_I3_J,axiom,
! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size_gen(3)
tff(fact_6530_mask__nat__positive__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% mask_nat_positive_iff
tff(fact_6531_zabs__less__one__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
<=> ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
tff(fact_6532_arctan__less__zero__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).
% arctan_less_zero_iff
tff(fact_6533_zero__less__arctan__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% zero_less_arctan_iff
tff(fact_6534_sum__abs,axiom,
! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A3))),groups4538972089207619220nt_int(aTP_Lamm_ep(F),A3)) ).
% sum_abs
tff(fact_6535_sum__abs,axiom,
! [F: fun_nat_real,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(F,A3))),groups6591440286371151544t_real(aTP_Lamm_eq(F),A3)) ).
% sum_abs
tff(fact_6536_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_6537_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_6538_sum__abs__ge__zero,axiom,
! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_ep(F),A3)) ).
% sum_abs_ge_zero
tff(fact_6539_sum__abs__ge__zero,axiom,
! [F: fun_nat_real,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_eq(F),A3)) ).
% sum_abs_ge_zero
tff(fact_6540_arctan__less__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% arctan_less_iff
tff(fact_6541_arctan__monotone,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).
% arctan_monotone
tff(fact_6542_less__eq__mask,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).
% less_eq_mask
tff(fact_6543_sum__mono,axiom,
! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),K5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).
% sum_mono
tff(fact_6544_sum__mono,axiom,
! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),K5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).
% sum_mono
tff(fact_6545_sum__mono,axiom,
! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),K5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).
% sum_mono
tff(fact_6546_sum__mono,axiom,
! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
( ! [I3: int] :
( aa_set_int_o(member_int(I3),K5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).
% sum_mono
tff(fact_6547_sum__mono,axiom,
! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),K5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).
% sum_mono
tff(fact_6548_sum__mono,axiom,
! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),K5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).
% sum_mono
tff(fact_6549_sum__mono,axiom,
! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
( ! [I3: int] :
( aa_set_int_o(member_int(I3),K5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).
% sum_mono
tff(fact_6550_sum__mono,axiom,
! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),K5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).
% sum_mono
tff(fact_6551_sum__mono,axiom,
! [K5: set_real,F: fun_real_int,G: fun_real_int] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),K5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).
% sum_mono
tff(fact_6552_sum__mono,axiom,
! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),K5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).
% sum_mono
tff(fact_6553_sum__distrib__left,axiom,
! [R2: int,F: fun_int_int,A3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_er(R2),F),A3) ).
% sum_distrib_left
tff(fact_6554_sum__distrib__left,axiom,
! [R2: complex,F: fun_complex_complex,A3: set_complex] : aa_complex_complex(times_times_complex(R2),groups7754918857620584856omplex(F,A3)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_es(R2),F),A3) ).
% sum_distrib_left
tff(fact_6555_sum__distrib__left,axiom,
! [R2: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_et(R2),F)),A3) ).
% sum_distrib_left
tff(fact_6556_sum__distrib__left,axiom,
! [R2: real,F: fun_nat_real,A3: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),groups6591440286371151544t_real(F,A3)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eu(R2),F),A3) ).
% sum_distrib_left
tff(fact_6557_sum__distrib__right,axiom,
! [F: fun_int_int,A3: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ev(F),R2),A3) ).
% sum_distrib_right
tff(fact_6558_sum__distrib__right,axiom,
! [F: fun_complex_complex,A3: set_complex,R2: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A3)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ew(F),R2),A3) ).
% sum_distrib_right
tff(fact_6559_sum__distrib__right,axiom,
! [F: fun_nat_nat,A3: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ex(F),R2)),A3) ).
% sum_distrib_right
tff(fact_6560_sum__distrib__right,axiom,
! [F: fun_nat_real,A3: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A3)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ey(F),R2),A3) ).
% sum_distrib_right
tff(fact_6561_sum__product,axiom,
! [F: fun_int_int,A3: set_int,G: fun_int_int,B3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,B3)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_fa(F),G),B3),A3) ).
% sum_product
tff(fact_6562_sum__product,axiom,
! [F: fun_complex_complex,A3: set_complex,G: fun_complex_complex,B3: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A3)),groups7754918857620584856omplex(G,B3)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_fc(F),G),B3),A3) ).
% sum_product
tff(fact_6563_sum__product,axiom,
! [F: fun_nat_nat,A3: set_nat,G: fun_nat_nat,B3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_fe(F),G),B3)),A3) ).
% sum_product
tff(fact_6564_sum__product,axiom,
! [F: fun_nat_real,A3: set_nat,G: fun_nat_real,B3: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A3)),groups6591440286371151544t_real(G,B3)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_fg(F),G),B3),A3) ).
% sum_product
tff(fact_6565_sum_Odistrib,axiom,
! [G: fun_int_int,H: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fh(G),H),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A3)),groups4538972089207619220nt_int(H,A3)) ).
% sum.distrib
tff(fact_6566_sum_Odistrib,axiom,
! [G: fun_complex_complex,H: fun_complex_complex,A3: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fi(G),H),A3) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A3)),groups7754918857620584856omplex(H,A3)) ).
% sum.distrib
tff(fact_6567_sum_Odistrib,axiom,
! [G: fun_nat_nat,H: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fj(G),H)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A3)) ).
% sum.distrib
tff(fact_6568_sum_Odistrib,axiom,
! [G: fun_nat_real,H: fun_nat_real,A3: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fk(G),H),A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A3)),groups6591440286371151544t_real(H,A3)) ).
% sum.distrib
tff(fact_6569_sum__nonpos,axiom,
! [A3: set_complex,F: fun_complex_rat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),zero_zero_rat) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),zero_zero_rat) ) ).
% sum_nonpos
tff(fact_6570_sum__nonpos,axiom,
! [A3: set_real,F: fun_real_rat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X4)),zero_zero_rat) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),zero_zero_rat) ) ).
% sum_nonpos
tff(fact_6571_sum__nonpos,axiom,
! [A3: set_nat,F: fun_nat_rat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),zero_zero_rat) ) ).
% sum_nonpos
tff(fact_6572_sum__nonpos,axiom,
! [A3: set_int,F: fun_int_rat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),zero_zero_rat) ) ).
% sum_nonpos
tff(fact_6573_sum__nonpos,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),zero_zero_nat) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A3)),zero_zero_nat) ) ).
% sum_nonpos
tff(fact_6574_sum__nonpos,axiom,
! [A3: set_real,F: fun_real_nat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),zero_zero_nat) ) ).
% sum_nonpos
tff(fact_6575_sum__nonpos,axiom,
! [A3: set_int,F: fun_int_nat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),zero_zero_nat) ) ).
% sum_nonpos
tff(fact_6576_sum__nonpos,axiom,
! [A3: set_complex,F: fun_complex_int] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),zero_zero_int) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A3)),zero_zero_int) ) ).
% sum_nonpos
tff(fact_6577_sum__nonpos,axiom,
! [A3: set_real,F: fun_real_int] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),zero_zero_int) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A3)),zero_zero_int) ) ).
% sum_nonpos
tff(fact_6578_sum__nonpos,axiom,
! [A3: set_nat,F: fun_nat_int] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),zero_zero_int) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A3)),zero_zero_int) ) ).
% sum_nonpos
tff(fact_6579_sum__nonneg,axiom,
! [A3: set_complex,F: fun_complex_rat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A3)) ) ).
% sum_nonneg
tff(fact_6580_sum__nonneg,axiom,
! [A3: set_real,F: fun_real_rat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A3)) ) ).
% sum_nonneg
tff(fact_6581_sum__nonneg,axiom,
! [A3: set_nat,F: fun_nat_rat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A3)) ) ).
% sum_nonneg
tff(fact_6582_sum__nonneg,axiom,
! [A3: set_int,F: fun_int_rat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3)) ) ).
% sum_nonneg
tff(fact_6583_sum__nonneg,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A3)) ) ).
% sum_nonneg
tff(fact_6584_sum__nonneg,axiom,
! [A3: set_real,F: fun_real_nat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A3)) ) ).
% sum_nonneg
tff(fact_6585_sum__nonneg,axiom,
! [A3: set_int,F: fun_int_nat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3)) ) ).
% sum_nonneg
tff(fact_6586_sum__nonneg,axiom,
! [A3: set_complex,F: fun_complex_int] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups5690904116761175830ex_int(F,A3)) ) ).
% sum_nonneg
tff(fact_6587_sum__nonneg,axiom,
! [A3: set_real,F: fun_real_int] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A3)) ) ).
% sum_nonneg
tff(fact_6588_sum__nonneg,axiom,
! [A3: set_nat,F: fun_nat_int] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups3539618377306564664at_int(F,A3)) ) ).
% sum_nonneg
tff(fact_6589_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) )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_set_real_o(finite_finite_real,I5)
=> ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6590_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) )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
=> ( aa_set_nat_o(member_nat(I),I5)
=> ( aa_set_nat_o(finite_finite_nat,I5)
=> ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6591_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) )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
=> ( aa_set_int_o(member_int(I),I5)
=> ( aa_set_int_o(finite_finite_int,I5)
=> ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6592_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) )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6593_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) )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_set_real_o(finite_finite_real,I5)
=> ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6594_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) )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
=> ( aa_set_int_o(member_int(I),I5)
=> ( aa_set_int_o(finite_finite_int,I5)
=> ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6595_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) )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6596_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) )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_set_real_o(finite_finite_real,I5)
=> ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6597_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) )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
=> ( aa_set_nat_o(member_nat(I),I5)
=> ( aa_set_nat_o(finite_finite_nat,I5)
=> ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6598_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) )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) )
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_complex_int(F,I) = aa_complex_int(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6599_abs__zmult__eq__1,axiom,
! [Mb: int,Nb: int] :
( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
=> ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).
% abs_zmult_eq_1
tff(fact_6600_not__mask__negative__int,axiom,
! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).
% not_mask_negative_int
tff(fact_6601_sum__nonneg__eq__0__iff,axiom,
! [A3: set_real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
=> ( ( groups1300246762558778688al_rat(F,A3) = zero_zero_rat )
<=> ! [X: real] :
( aa_set_real_o(member_real(X),A3)
=> ( aa_real_rat(F,X) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6602_sum__nonneg__eq__0__iff,axiom,
! [A3: set_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
=> ( ( groups2906978787729119204at_rat(F,A3) = zero_zero_rat )
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
=> ( aa_nat_rat(F,X) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6603_sum__nonneg__eq__0__iff,axiom,
! [A3: set_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
=> ( ( groups3906332499630173760nt_rat(F,A3) = zero_zero_rat )
<=> ! [X: int] :
( aa_set_int_o(member_int(X),A3)
=> ( aa_int_rat(F,X) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6604_sum__nonneg__eq__0__iff,axiom,
! [A3: set_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
=> ( ( groups5058264527183730370ex_rat(F,A3) = zero_zero_rat )
<=> ! [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
=> ( aa_complex_rat(F,X) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6605_sum__nonneg__eq__0__iff,axiom,
! [A3: set_real,F: fun_real_nat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
=> ( ( groups1935376822645274424al_nat(F,A3) = zero_zero_nat )
<=> ! [X: real] :
( aa_set_real_o(member_real(X),A3)
=> ( aa_real_nat(F,X) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6606_sum__nonneg__eq__0__iff,axiom,
! [A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
=> ( ( groups4541462559716669496nt_nat(F,A3) = zero_zero_nat )
<=> ! [X: int] :
( aa_set_int_o(member_int(X),A3)
=> ( aa_int_nat(F,X) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6607_sum__nonneg__eq__0__iff,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
=> ( ( groups5693394587270226106ex_nat(F,A3) = zero_zero_nat )
<=> ! [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
=> ( aa_complex_nat(F,X) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6608_sum__nonneg__eq__0__iff,axiom,
! [A3: set_real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)) )
=> ( ( groups1932886352136224148al_int(F,A3) = zero_zero_int )
<=> ! [X: real] :
( aa_set_real_o(member_real(X),A3)
=> ( aa_real_int(F,X) = zero_zero_int ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6609_sum__nonneg__eq__0__iff,axiom,
! [A3: set_nat,F: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
=> ( ( groups3539618377306564664at_int(F,A3) = zero_zero_int )
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
=> ( aa_nat_int(F,X) = zero_zero_int ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6610_sum__nonneg__eq__0__iff,axiom,
! [A3: set_complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4)) )
=> ( ( groups5690904116761175830ex_int(F,A3) = zero_zero_int )
<=> ! [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
=> ( aa_complex_int(F,X) = zero_zero_int ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6611_sum__le__included,axiom,
! [S2: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( aa_set_nat_o(finite_finite_nat,Ta)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4)) )
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S2)
=> ? [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),Ta)
& ( aa_nat_nat(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6612_sum__le__included,axiom,
! [S2: set_nat,Ta: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( aa_set_int_o(finite_finite_int,Ta)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4)) )
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S2)
=> ? [Xa: int] :
( aa_set_int_o(member_int(Xa),Ta)
& ( aa_int_nat(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_int_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6613_sum__le__included,axiom,
! [S2: set_nat,Ta: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4)) )
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S2)
=> ? [Xa: complex] :
( aa_set_complex_o(member_complex(Xa),Ta)
& ( aa_complex_nat(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_complex_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6614_sum__le__included,axiom,
! [S2: set_int,Ta: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,S2)
=> ( aa_set_nat_o(finite_finite_nat,Ta)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4)) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S2)
=> ? [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),Ta)
& ( aa_nat_int(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),aa_nat_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6615_sum__le__included,axiom,
! [S2: set_int,Ta: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,S2)
=> ( aa_set_int_o(finite_finite_int,Ta)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4)) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S2)
=> ? [Xa: int] :
( aa_set_int_o(member_int(Xa),Ta)
& ( aa_int_int(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S2)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6616_sum__le__included,axiom,
! [S2: set_int,Ta: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,S2)
=> ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4)) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S2)
=> ? [Xa: complex] :
( aa_set_complex_o(member_complex(Xa),Ta)
& ( aa_complex_int(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),aa_complex_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6617_sum__le__included,axiom,
! [S2: set_complex,Ta: set_nat,G: fun_nat_rat,I: fun_nat_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( aa_set_nat_o(finite_finite_nat,Ta)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4)) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S2)
=> ? [Xa: nat] :
( aa_set_nat_o(member_nat(Xa),Ta)
& ( aa_nat_complex(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_nat_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6618_sum__le__included,axiom,
! [S2: set_complex,Ta: set_int,G: fun_int_rat,I: fun_int_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( aa_set_int_o(finite_finite_int,Ta)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4)) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S2)
=> ? [Xa: int] :
( aa_set_int_o(member_int(Xa),Ta)
& ( aa_int_complex(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_int_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S2)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6619_sum__le__included,axiom,
! [S2: set_complex,Ta: set_complex,G: fun_complex_rat,I: fun_complex_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),Ta)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4)) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S2)
=> ? [Xa: complex] :
( aa_set_complex_o(member_complex(Xa),Ta)
& ( aa_complex_complex(I,Xa) = X4 )
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,Xa)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6620_sum__le__included,axiom,
! [S2: set_int,Ta: set_int,G: fun_int_nat,I: fun_int_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,S2)
=> ( aa_set_int_o(finite_finite_int,Ta)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),Ta)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(G,X4)) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S2)
=> ? [Xa: int] :
( aa_set_int_o(member_int(Xa),Ta)
& ( aa_int_int(I,Xa) = X4 )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,Xa)) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,S2)),groups4541462559716669496nt_nat(G,Ta)) ) ) ) ) ).
% sum_le_included
tff(fact_6621_sum__strict__mono__ex1,axiom,
! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4)) )
=> ( ? [X3: nat] :
( aa_set_nat_o(member_nat(X3),A3)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6622_sum__strict__mono__ex1,axiom,
! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4)) )
=> ( ? [X3: int] :
( aa_set_int_o(member_int(X3),A3)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6623_sum__strict__mono__ex1,axiom,
! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4)) )
=> ( ? [X3: complex] :
( aa_set_complex_o(member_complex(X3),A3)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6624_sum__strict__mono__ex1,axiom,
! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4)) )
=> ( ? [X3: int] :
( aa_set_int_o(member_int(X3),A3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6625_sum__strict__mono__ex1,axiom,
! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4)) )
=> ( ? [X3: complex] :
( aa_set_complex_o(member_complex(X3),A3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6626_sum__strict__mono__ex1,axiom,
! [A3: set_nat,F: fun_nat_int,G: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4)) )
=> ( ? [X3: nat] :
( aa_set_nat_o(member_nat(X3),A3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6627_sum__strict__mono__ex1,axiom,
! [A3: set_complex,F: fun_complex_int,G: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4)) )
=> ( ? [X3: complex] :
( aa_set_complex_o(member_complex(X3),A3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X3)),aa_complex_int(G,X3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6628_sum__strict__mono__ex1,axiom,
! [A3: set_int,F: fun_int_real,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4)) )
=> ( ? [X3: int] :
( aa_set_int_o(member_int(X3),A3)
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6629_sum__strict__mono__ex1,axiom,
! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4)) )
=> ( ? [X3: complex] :
( aa_set_complex_o(member_complex(X3),A3)
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6630_sum__strict__mono__ex1,axiom,
! [A3: set_int,F: fun_int_int,G: fun_int_int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4)) )
=> ( ? [X3: int] :
( aa_set_int_o(member_int(X3),A3)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,A3)) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6631_sum_Orelated,axiom,
! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
& aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
=> ( aa_set_nat_o(finite_finite_nat,S3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4)) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,groups2073611262835488442omplex(H,S3)),groups2073611262835488442omplex(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6632_sum_Orelated,axiom,
! [R: fun_co1148497455491383662plex_o,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
& aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X4)),aa_int_complex(G,X4)) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,groups3049146728041665814omplex(H,S3)),groups3049146728041665814omplex(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6633_sum_Orelated,axiom,
! [R: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
& aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
=> aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X4)),aa_int_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6634_sum_Orelated,axiom,
! [R: fun_real_fun_real_o,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
& aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
=> aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X4)),aa_complex_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6635_sum_Orelated,axiom,
! [R: fun_rat_fun_rat_o,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
& aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
=> ( aa_set_nat_o(finite_finite_nat,S3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6636_sum_Orelated,axiom,
! [R: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
& aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X4)),aa_int_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(R,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6637_sum_Orelated,axiom,
! [R: fun_rat_fun_rat_o,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
& aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6638_sum_Orelated,axiom,
! [R: fun_nat_fun_nat_o,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
=> ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
& aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
=> aa_nat_o(aa_nat_fun_nat_o(R,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)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X4)),aa_int_nat(G,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(R,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6639_sum_Orelated,axiom,
! [R: fun_nat_fun_nat_o,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
=> ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
& aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
=> aa_nat_o(aa_nat_fun_nat_o(R,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)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6640_sum_Orelated,axiom,
! [R: fun_int_fun_int_o,S3: set_nat,H: fun_nat_int,G: fun_nat_int] :
( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
=> ( ! [X15: int,Y15: int,X22: int,Y23: int] :
( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
& aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
=> aa_int_o(aa_int_fun_int_o(R,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)) )
=> ( aa_set_nat_o(finite_finite_nat,S3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
=> aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X4)),aa_nat_int(G,X4)) )
=> aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S3)),groups3539618377306564664at_int(G,S3)) ) ) ) ) ).
% sum.related
tff(fact_6641_sum__strict__mono,axiom,
! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ( A3 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6642_sum__strict__mono,axiom,
! [A3: set_int,F: fun_int_real,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( A3 != bot_bot_set_int )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6643_sum__strict__mono,axiom,
! [A3: set_real,F: fun_real_real,G: fun_real_real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ( A3 != bot_bot_set_real )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6644_sum__strict__mono,axiom,
! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ( A3 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6645_sum__strict__mono,axiom,
! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6646_sum__strict__mono,axiom,
! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( A3 != bot_bot_set_int )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6647_sum__strict__mono,axiom,
! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ( A3 != bot_bot_set_real )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6648_sum__strict__mono,axiom,
! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ( A3 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6649_sum__strict__mono,axiom,
! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( A3 != bot_bot_set_int )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6650_sum__strict__mono,axiom,
! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ( A3 != bot_bot_set_real )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(G,A3)) ) ) ) ).
% sum_strict_mono
tff(fact_6651_zabs__def,axiom,
! [I: int] :
aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).
% zabs_def
tff(fact_6652_sum__nonneg__0,axiom,
! [S2: set_real,F: fun_real_rat,I: real] :
( aa_set_real_o(finite_finite_real,S2)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
=> ( ( groups1300246762558778688al_rat(F,S2) = zero_zero_rat )
=> ( aa_set_real_o(member_real(I),S2)
=> ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6653_sum__nonneg__0,axiom,
! [S2: set_nat,F: fun_nat_rat,I: nat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
=> ( ( groups2906978787729119204at_rat(F,S2) = zero_zero_rat )
=> ( aa_set_nat_o(member_nat(I),S2)
=> ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6654_sum__nonneg__0,axiom,
! [S2: set_int,F: fun_int_rat,I: int] :
( aa_set_int_o(finite_finite_int,S2)
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
=> ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
=> ( aa_set_int_o(member_int(I),S2)
=> ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6655_sum__nonneg__0,axiom,
! [S2: set_complex,F: fun_complex_rat,I: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
=> ( ( groups5058264527183730370ex_rat(F,S2) = zero_zero_rat )
=> ( aa_set_complex_o(member_complex(I),S2)
=> ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6656_sum__nonneg__0,axiom,
! [S2: set_real,F: fun_real_nat,I: real] :
( aa_set_real_o(finite_finite_real,S2)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
=> ( ( groups1935376822645274424al_nat(F,S2) = zero_zero_nat )
=> ( aa_set_real_o(member_real(I),S2)
=> ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6657_sum__nonneg__0,axiom,
! [S2: set_int,F: fun_int_nat,I: int] :
( aa_set_int_o(finite_finite_int,S2)
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
=> ( ( groups4541462559716669496nt_nat(F,S2) = zero_zero_nat )
=> ( aa_set_int_o(member_int(I),S2)
=> ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6658_sum__nonneg__0,axiom,
! [S2: set_complex,F: fun_complex_nat,I: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
=> ( ( groups5693394587270226106ex_nat(F,S2) = zero_zero_nat )
=> ( aa_set_complex_o(member_complex(I),S2)
=> ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6659_sum__nonneg__0,axiom,
! [S2: set_real,F: fun_real_int,I: real] :
( aa_set_real_o(finite_finite_real,S2)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3)) )
=> ( ( groups1932886352136224148al_int(F,S2) = zero_zero_int )
=> ( aa_set_real_o(member_real(I),S2)
=> ( aa_real_int(F,I) = zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6660_sum__nonneg__0,axiom,
! [S2: set_nat,F: fun_nat_int,I: nat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I3)) )
=> ( ( groups3539618377306564664at_int(F,S2) = zero_zero_int )
=> ( aa_set_nat_o(member_nat(I),S2)
=> ( aa_nat_int(F,I) = zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6661_sum__nonneg__0,axiom,
! [S2: set_complex,F: fun_complex_int,I: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I3)) )
=> ( ( groups5690904116761175830ex_int(F,S2) = zero_zero_int )
=> ( aa_set_complex_o(member_complex(I),S2)
=> ( aa_complex_int(F,I) = zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6662_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: fun_real_rat,B3: rat,I: real] :
( aa_set_real_o(finite_finite_real,S2)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
=> ( ( groups1300246762558778688al_rat(F,S2) = B3 )
=> ( aa_set_real_o(member_real(I),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6663_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: fun_nat_rat,B3: rat,I: nat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
=> ( ( groups2906978787729119204at_rat(F,S2) = B3 )
=> ( aa_set_nat_o(member_nat(I),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6664_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: fun_int_rat,B3: rat,I: int] :
( aa_set_int_o(finite_finite_int,S2)
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
=> ( ( groups3906332499630173760nt_rat(F,S2) = B3 )
=> ( aa_set_int_o(member_int(I),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6665_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: fun_complex_rat,B3: rat,I: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
=> ( ( groups5058264527183730370ex_rat(F,S2) = B3 )
=> ( aa_set_complex_o(member_complex(I),S2)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6666_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: fun_real_nat,B3: nat,I: real] :
( aa_set_real_o(finite_finite_real,S2)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
=> ( ( groups1935376822645274424al_nat(F,S2) = B3 )
=> ( aa_set_real_o(member_real(I),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6667_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: fun_int_nat,B3: nat,I: int] :
( aa_set_int_o(finite_finite_int,S2)
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
=> ( ( groups4541462559716669496nt_nat(F,S2) = B3 )
=> ( aa_set_int_o(member_int(I),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6668_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: fun_complex_nat,B3: nat,I: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
=> ( ( groups5693394587270226106ex_nat(F,S2) = B3 )
=> ( aa_set_complex_o(member_complex(I),S2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6669_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: fun_real_int,B3: int,I: real] :
( aa_set_real_o(finite_finite_real,S2)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3)) )
=> ( ( groups1932886352136224148al_int(F,S2) = B3 )
=> ( aa_set_real_o(member_real(I),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6670_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: fun_nat_int,B3: int,I: nat] :
( aa_set_nat_o(finite_finite_nat,S2)
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I3)) )
=> ( ( groups3539618377306564664at_int(F,S2) = B3 )
=> ( aa_set_nat_o(member_nat(I),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6671_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: fun_complex_int,B3: int,I: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S2)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I3)) )
=> ( ( groups5690904116761175830ex_int(F,S2) = B3 )
=> ( aa_set_complex_o(member_complex(I),S2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I)),B3) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6672_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L)) ) ).
% abs_mod_less
tff(fact_6673_less__mask,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).
% less_mask
tff(fact_6674_sum__pos2,axiom,
! [I5: set_real,I: real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,I5)
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6675_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( aa_set_nat_o(member_nat(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6676_sum__pos2,axiom,
! [I5: set_int,I: int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,I5)
=> ( aa_set_int_o(member_int(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6677_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6678_sum__pos2,axiom,
! [I5: set_real,I: real,F: fun_real_nat] :
( aa_set_real_o(finite_finite_real,I5)
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I))
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6679_sum__pos2,axiom,
! [I5: set_int,I: int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,I5)
=> ( aa_set_int_o(member_int(I),I5)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6680_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I))
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6681_sum__pos2,axiom,
! [I5: set_real,I: real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,I5)
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,I))
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1932886352136224148al_int(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6682_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( aa_set_nat_o(member_nat(I),I5)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups3539618377306564664at_int(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6683_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,I))
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups5690904116761175830ex_int(F,I5)) ) ) ) ) ).
% sum_pos2
tff(fact_6684_sum__pos,axiom,
! [I5: set_complex,F: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6685_sum__pos,axiom,
! [I5: set_int,F: fun_int_real] :
( aa_set_int_o(finite_finite_int,I5)
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6686_sum__pos,axiom,
! [I5: set_real,F: fun_real_real] :
( aa_set_real_o(finite_finite_real,I5)
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6687_sum__pos,axiom,
! [I5: set_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6688_sum__pos,axiom,
! [I5: set_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6689_sum__pos,axiom,
! [I5: set_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,I5)
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6690_sum__pos,axiom,
! [I5: set_real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,I5)
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6691_sum__pos,axiom,
! [I5: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6692_sum__pos,axiom,
! [I5: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,I5)
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6693_sum__pos,axiom,
! [I5: set_real,F: fun_real_nat] :
( aa_set_real_o(finite_finite_real,I5)
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I3)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ).
% sum_pos
tff(fact_6694_sum_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_real] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups8778361861064173332t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A3,B3))),groups8778361861064173332t_real(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6695_sum_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_real] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5808333547571424918x_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,B3))),groups5808333547571424918x_real(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6696_sum_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_rat] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups3906332499630173760nt_rat(G,A3) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,B3))),groups3906332499630173760nt_rat(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6697_sum_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_rat] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5058264527183730370ex_rat(G,A3) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,B3))),groups5058264527183730370ex_rat(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6698_sum_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_nat] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups4541462559716669496nt_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,B3))),groups4541462559716669496nt_nat(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6699_sum_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_nat] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5693394587270226106ex_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A3,B3))),groups5693394587270226106ex_nat(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6700_sum_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_int] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5690904116761175830ex_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A3,B3))),groups5690904116761175830ex_int(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6701_sum_Osubset__diff,axiom,
! [B3: set_nat,A3: set_nat,G: fun_nat_rat] :
( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups2906978787729119204at_rat(G,A3) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A3,B3))),groups2906978787729119204at_rat(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6702_sum_Osubset__diff,axiom,
! [B3: set_nat,A3: set_nat,G: fun_nat_int] :
( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups3539618377306564664at_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A3,B3))),groups3539618377306564664at_int(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6703_sum_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_int] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups4538972089207619220nt_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A3,B3))),groups4538972089207619220nt_int(G,B3)) ) ) ) ).
% sum.subset_diff
tff(fact_6704_zdvd__mult__cancel1,axiom,
! [Mb: int,Nb: int] :
( ( Mb != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)),Mb)
<=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).
% zdvd_mult_cancel1
tff(fact_6705_sum__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6706_sum__mono2,axiom,
! [B3: set_int,A3: set_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6707_sum__mono2,axiom,
! [B3: set_complex,A3: set_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6708_sum__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_nat] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6709_sum__mono2,axiom,
! [B3: set_int,A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(B3,A3))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6710_sum__mono2,axiom,
! [B3: set_complex,A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6711_sum__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A3)),groups1932886352136224148al_int(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6712_sum__mono2,axiom,
! [B3: set_complex,A3: set_complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6713_sum__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_real] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6714_sum__mono2,axiom,
! [B3: set_int,A3: set_int,F: fun_int_real] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(B3,A3))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,B4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B3)) ) ) ) ).
% sum_mono2
tff(fact_6715_even__add__abs__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L)))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).
% even_add_abs_iff
tff(fact_6716_even__abs__add__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).
% even_abs_add_iff
tff(fact_6717_num_Osize__gen_I1_J,axiom,
aa_num_nat(size_num,one) = zero_zero_nat ).
% num.size_gen(1)
tff(fact_6718_sum__strict__mono2,axiom,
! [B3: set_real,A3: set_real,B: real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( aa_set_real_o(member_real(B),minus_minus_set_real(B3,A3))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),B3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6719_sum__strict__mono2,axiom,
! [B3: set_int,A3: set_int,B: int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( aa_set_int_o(member_int(B),minus_minus_set_int(B3,A3))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,B))
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),B3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6720_sum__strict__mono2,axiom,
! [B3: set_complex,A3: set_complex,B: complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B3,A3))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),B3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6721_sum__strict__mono2,axiom,
! [B3: set_real,A3: set_real,B: real,F: fun_real_nat] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( aa_set_real_o(member_real(B),minus_minus_set_real(B3,A3))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),B3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6722_sum__strict__mono2,axiom,
! [B3: set_int,A3: set_int,B: int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( aa_set_int_o(member_int(B),minus_minus_set_int(B3,A3))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,B))
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),B3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6723_sum__strict__mono2,axiom,
! [B3: set_complex,A3: set_complex,B: complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B3,A3))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),B3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6724_sum__strict__mono2,axiom,
! [B3: set_real,A3: set_real,B: real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( aa_set_real_o(member_real(B),minus_minus_set_real(B3,A3))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),B3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A3)),groups1932886352136224148al_int(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6725_sum__strict__mono2,axiom,
! [B3: set_complex,A3: set_complex,B: complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B3,A3))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,B))
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),B3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6726_sum__strict__mono2,axiom,
! [B3: set_real,A3: set_real,B: real,F: fun_real_real] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( aa_set_real_o(member_real(B),minus_minus_set_real(B3,A3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),B3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6727_sum__strict__mono2,axiom,
! [B3: set_int,A3: set_int,B: int,F: fun_int_real] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( aa_set_int_o(member_int(B),minus_minus_set_int(B3,A3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,B))
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),B3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B3)) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6728_Suc__mask__eq__exp,axiom,
! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).
% Suc_mask_eq_exp
tff(fact_6729_mask__nat__less__exp,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% mask_nat_less_exp
tff(fact_6730_convex__sum__bound__le,axiom,
! [I5: set_complex,Xb: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xb,I3)) )
=> ( ( groups6621422865394947399nteger(Xb,I5) = one_one_Code_integer )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I3),B))),Delta) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_fl(Xb),A),I5),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6731_convex__sum__bound__le,axiom,
! [I5: set_real,Xb: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I3)) )
=> ( ( groups7713935264441627589nteger(Xb,I5) = one_one_Code_integer )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_fm(Xb),A),I5),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6732_convex__sum__bound__le,axiom,
! [I5: set_nat,Xb: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I3)) )
=> ( ( groups7501900531339628137nteger(Xb,I5) = one_one_Code_integer )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_fn(Xb),A),I5),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6733_convex__sum__bound__le,axiom,
! [I5: set_int,Xb: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I3)) )
=> ( ( groups7873554091576472773nteger(Xb,I5) = one_one_Code_integer )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_fo(Xb),A),I5),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6734_convex__sum__bound__le,axiom,
! [I5: set_complex,Xb: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Xb,I3)) )
=> ( ( groups5058264527183730370ex_rat(Xb,I5) = one_one_rat )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I3)),B))),Delta) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fp(Xb),A),I5)),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6735_convex__sum__bound__le,axiom,
! [I5: set_real,Xb: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I3)) )
=> ( ( groups1300246762558778688al_rat(Xb,I5) = one_one_rat )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I3)),B))),Delta) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fq(Xb),A),I5)),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6736_convex__sum__bound__le,axiom,
! [I5: set_nat,Xb: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I3)) )
=> ( ( groups2906978787729119204at_rat(Xb,I5) = one_one_rat )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I3)),B))),Delta) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fr(Xb),A),I5)),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6737_convex__sum__bound__le,axiom,
! [I5: set_int,Xb: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Xb,I3)) )
=> ( ( groups3906332499630173760nt_rat(Xb,I5) = one_one_rat )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(A,I3)),B))),Delta) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fs(Xb),A),I5)),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6738_convex__sum__bound__le,axiom,
! [I5: set_complex,Xb: fun_complex_int,A: fun_complex_int,B: int,Delta: int] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(Xb,I3)) )
=> ( ( groups5690904116761175830ex_int(Xb,I5) = one_one_int )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_complex_int(A,I3)),B))),Delta) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_ft(Xb),A),I5)),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6739_convex__sum__bound__le,axiom,
! [I5: set_real,Xb: fun_real_int,A: fun_real_int,B: int,Delta: int] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(Xb,I3)) )
=> ( ( groups1932886352136224148al_int(Xb,I5) = one_one_int )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(A,I3)),B))),Delta) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_fu(Xb),A),I5)),B))),Delta) ) ) ) ).
% convex_sum_bound_le
tff(fact_6740_nat__intermed__int__val,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_int,K: int] :
( ! [I3: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb) )
=> aa_int_o(aa_int_fun_int_o(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,I3))),aa_nat_int(F,I3)))),one_one_int) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
=> ? [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
& ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).
% nat_intermed_int_val
tff(fact_6741_incr__lemma,axiom,
! [D: int,Z: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),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,Xb),Z))),one_one_int)),D))) ) ).
% incr_lemma
tff(fact_6742_decr__lemma,axiom,
! [D: int,Xb: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),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,Xb),Z))),one_one_int)),D))),Z) ) ).
% decr_lemma
tff(fact_6743_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb))
<=> ( Nb = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_6744_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
<=> ( Nb = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_6745_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))
<=> ( Nb = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_6746_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_6747_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_6748_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B = aa_rat_rat(plus_plus_rat(B),A) )
<=> ( A = zero_zero_rat ) ) ).
% add_0_iff
tff(fact_6749_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_6750_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_6751_crossproduct__eq,axiom,
! [W: real,Y: real,Xb: 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,Xb),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,Xb),Y)) )
<=> ( ( W = Xb )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6752_crossproduct__eq,axiom,
! [W: rat,Y: rat,Xb: rat,Z: rat] :
( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Xb),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(Xb),Y)) )
<=> ( ( W = Xb )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6753_crossproduct__eq,axiom,
! [W: nat,Y: nat,Xb: 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,Xb),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,Xb),Y)) )
<=> ( ( W = Xb )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6754_crossproduct__eq,axiom,
! [W: int,Y: int,Xb: 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,Xb),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,Xb),Y)) )
<=> ( ( W = Xb )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6755_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_6756_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( A != B )
& ( C != D ) )
<=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).
% crossproduct_noteq
tff(fact_6757_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_6758_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_6759_mask__nat__def,axiom,
! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = 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))),Nb)),one_one_nat) ).
% mask_nat_def
tff(fact_6760_mask__half__int,axiom,
! [Nb: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,Nb)),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,Nb),one_one_nat)) ).
% mask_half_int
tff(fact_6761_mask__int__def,axiom,
! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).
% mask_int_def
tff(fact_6762_nat__ivt__aux,axiom,
! [Nb: nat,F: fun_nat_int,K: int] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
=> aa_int_o(aa_int_fun_int_o(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,I3))),aa_nat_int(F,I3)))),one_one_int) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
=> ? [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
& ( aa_nat_int(F,I3) = K ) ) ) ) ) ).
% nat_ivt_aux
tff(fact_6763_mask__eq__exp__minus__1,axiom,
! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = 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))),Nb)),one_one_nat) ).
% mask_eq_exp_minus_1
tff(fact_6764_mask__eq__exp__minus__1,axiom,
! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).
% mask_eq_exp_minus_1
tff(fact_6765_nat0__intermed__int__val,axiom,
! [Nb: nat,F: fun_nat_int,K: int] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
=> aa_int_o(aa_int_fun_int_o(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,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
=> ? [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
& ( aa_nat_int(F,I3) = K ) ) ) ) ) ).
% nat0_intermed_int_val
tff(fact_6766_arctan__add,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)))) ) ) ) ).
% arctan_add
tff(fact_6767_num_Osize__gen_I2_J,axiom,
! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size_gen(2)
tff(fact_6768_take__bit__rec,axiom,
! [Nb: nat,A: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_6769_take__bit__rec,axiom,
! [Nb: nat,A: int] :
aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = zero_zero_nat,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,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_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(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).
% take_bit_rec
tff(fact_6770_tanh__real__altdef,axiom,
! [Xb: real] : aa_real_real(tanh_real,Xb) = 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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))) ).
% tanh_real_altdef
tff(fact_6771_and__int__unfold,axiom,
! [K: int,L: int] :
aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
( ( K = zero_zero_int )
| ( L = zero_zero_int ) ),
zero_zero_int,
$ite(
K = aa_int_int(uminus_uminus_int,one_one_int),
L,
$ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,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(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).
% and_int_unfold
tff(fact_6772_power__numeral,axiom,
! [K: num,L: num] : power_8040749407984259932d_enat(numera1916890842035813515d_enat(K),aa_num_nat(numeral_numeral_nat,L)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).
% power_numeral
tff(fact_6773_power__numeral,axiom,
! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).
% power_numeral
tff(fact_6774_power__numeral,axiom,
! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).
% power_numeral
tff(fact_6775_power__numeral,axiom,
! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).
% power_numeral
tff(fact_6776_power__numeral,axiom,
! [K: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).
% power_numeral
tff(fact_6777_or__int__unfold,axiom,
! [K: int,L: int] :
aa_int_int(bit_se1409905431419307370or_int(K),L) = $ite(
( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
| ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
aa_int_int(uminus_uminus_int,one_one_int),
$ite(
K = zero_zero_int,
L,
$ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_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(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).
% or_int_unfold
tff(fact_6778_arctan__half,axiom,
! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).
% arctan_half
tff(fact_6779_real__sqrt__less__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% real_sqrt_less_iff
tff(fact_6780_exp__less__cancel__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% exp_less_cancel_iff
tff(fact_6781_exp__less__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ).
% exp_less_mono
tff(fact_6782_exp__zero,axiom,
aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).
% exp_zero
tff(fact_6783_exp__zero,axiom,
aa_real_real(exp_real,zero_zero_real) = one_one_real ).
% exp_zero
tff(fact_6784_take__bit__Suc__1,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),one_one_nat) = one_one_nat ).
% take_bit_Suc_1
tff(fact_6785_take__bit__Suc__1,axiom,
! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).
% take_bit_Suc_1
tff(fact_6786_and_Oleft__neutral,axiom,
! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).
% and.left_neutral
tff(fact_6787_and_Oleft__neutral,axiom,
! [A: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).
% and.left_neutral
tff(fact_6788_and_Oright__neutral,axiom,
! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).
% and.right_neutral
tff(fact_6789_and_Oright__neutral,axiom,
! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).
% and.right_neutral
tff(fact_6790_bit_Oconj__one__right,axiom,
! [Xb: code_integer] : bit_se3949692690581998587nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Xb ).
% bit.conj_one_right
tff(fact_6791_bit_Oconj__one__right,axiom,
! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(Xb),aa_int_int(uminus_uminus_int,one_one_int)) = Xb ).
% bit.conj_one_right
tff(fact_6792_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_6793_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_6794_bit_Odisj__one__left,axiom,
! [Xb: code_integer] : bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.disj_one_left
tff(fact_6795_bit_Odisj__one__left,axiom,
! [Xb: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,one_one_int)),Xb) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.disj_one_left
tff(fact_6796_bit_Odisj__one__right,axiom,
! [Xb: code_integer] : bit_se1080825931792720795nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.disj_one_right
tff(fact_6797_bit_Odisj__one__right,axiom,
! [Xb: int] : aa_int_int(bit_se1409905431419307370or_int(Xb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.disj_one_right
tff(fact_6798_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).
% real_sqrt_gt_0_iff
tff(fact_6799_real__sqrt__lt__0__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).
% real_sqrt_lt_0_iff
tff(fact_6800_real__sqrt__lt__1__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).
% real_sqrt_lt_1_iff
tff(fact_6801_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).
% real_sqrt_gt_1_iff
tff(fact_6802_and__negative__int__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).
% and_negative_int_iff
tff(fact_6803_real__sqrt__abs2,axiom,
! [Xb: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).
% real_sqrt_abs2
tff(fact_6804_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_6805_or__negative__int__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
| aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).
% or_negative_int_iff
tff(fact_6806_take__bit__of__1__eq__0__iff,axiom,
! [Nb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_zero_nat )
<=> ( Nb = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_6807_take__bit__of__1__eq__0__iff,axiom,
! [Nb: nat] :
( ( aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = zero_zero_int )
<=> ( Nb = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_6808_and__numerals_I8_J,axiom,
! [Xb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = one_one_int ).
% and_numerals(8)
tff(fact_6809_and__numerals_I8_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),one_one_nat) = one_one_nat ).
% and_numerals(8)
tff(fact_6810_and__numerals_I2_J,axiom,
! [Y: num] : aa_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_6811_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_6812_or__numerals_I8_J,axiom,
! [Xb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).
% or_numerals(8)
tff(fact_6813_or__numerals_I8_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).
% or_numerals(8)
tff(fact_6814_or__numerals_I2_J,axiom,
! [Y: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).
% or_numerals(2)
tff(fact_6815_or__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).
% or_numerals(2)
tff(fact_6816_real__sqrt__four,axiom,
aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).
% real_sqrt_four
tff(fact_6817_exp__less__one__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).
% exp_less_one_iff
tff(fact_6818_one__less__exp__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% one_less_exp_iff
tff(fact_6819_take__bit__minus__one__eq__mask,axiom,
! [Nb: nat] : bit_se1745604003318907178nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb) ).
% take_bit_minus_one_eq_mask
tff(fact_6820_take__bit__minus__one__eq__mask,axiom,
! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).
% take_bit_minus_one_eq_mask
tff(fact_6821_exp__ln__iff,axiom,
! [Xb: real] :
( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb )
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% exp_ln_iff
tff(fact_6822_exp__ln,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb ) ) ).
% exp_ln
tff(fact_6823_take__bit__of__Suc__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% take_bit_of_Suc_0
tff(fact_6824_and__numerals_I5_J,axiom,
! [Xb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = zero_zero_int ).
% and_numerals(5)
tff(fact_6825_and__numerals_I5_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),one_one_nat) = zero_zero_nat ).
% and_numerals(5)
tff(fact_6826_and__numerals_I1_J,axiom,
! [Y: num] : aa_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_6827_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_6828_and__numerals_I3_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),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(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).
% and_numerals(3)
tff(fact_6829_and__numerals_I3_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).
% and_numerals(3)
tff(fact_6830_or__numerals_I3_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).
% or_numerals(3)
tff(fact_6831_or__numerals_I3_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).
% or_numerals(3)
tff(fact_6832_sum_Ocl__ivl__Suc,axiom,
! [G: fun_nat_complex,Mb: nat,Nb: nat] :
groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).
% sum.cl_ivl_Suc
tff(fact_6833_sum_Ocl__ivl__Suc,axiom,
! [G: fun_nat_rat,Mb: nat,Nb: nat] :
groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).
% sum.cl_ivl_Suc
tff(fact_6834_sum_Ocl__ivl__Suc,axiom,
! [G: fun_nat_int,Mb: nat,Nb: nat] :
groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).
% sum.cl_ivl_Suc
tff(fact_6835_sum_Ocl__ivl__Suc,axiom,
! [G: fun_nat_nat,Mb: nat,Nb: nat] :
aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,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(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).
% sum.cl_ivl_Suc
tff(fact_6836_sum_Ocl__ivl__Suc,axiom,
! [G: fun_nat_real,Mb: nat,Nb: nat] :
groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).
% sum.cl_ivl_Suc
tff(fact_6837_or__numerals_I5_J,axiom,
! [Xb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).
% or_numerals(5)
tff(fact_6838_or__numerals_I5_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).
% or_numerals(5)
tff(fact_6839_or__numerals_I1_J,axiom,
! [Y: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).
% or_numerals(1)
tff(fact_6840_or__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).
% or_numerals(1)
tff(fact_6841_take__bit__of__1,axiom,
! [Nb: nat] : bit_se1745604003318907178nteger(Nb,one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% take_bit_of_1
tff(fact_6842_take__bit__of__1,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% take_bit_of_1
tff(fact_6843_take__bit__of__1,axiom,
! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).
% take_bit_of_1
tff(fact_6844_sum__zero__power,axiom,
! [C: fun_nat_complex,A3: set_nat] :
groups2073611262835488442omplex(aTP_Lamm_fv(C),A3) = $ite(
( aa_set_nat_o(finite_finite_nat,A3)
& aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
aa_nat_complex(C,zero_zero_nat),
zero_zero_complex ) ).
% sum_zero_power
tff(fact_6845_sum__zero__power,axiom,
! [C: fun_nat_rat,A3: set_nat] :
groups2906978787729119204at_rat(aTP_Lamm_fw(C),A3) = $ite(
( aa_set_nat_o(finite_finite_nat,A3)
& aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
aa_nat_rat(C,zero_zero_nat),
zero_zero_rat ) ).
% sum_zero_power
tff(fact_6846_sum__zero__power,axiom,
! [C: fun_nat_real,A3: set_nat] :
groups6591440286371151544t_real(aTP_Lamm_fx(C),A3) = $ite(
( aa_set_nat_o(finite_finite_nat,A3)
& aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
aa_nat_real(C,zero_zero_nat),
zero_zero_real ) ).
% sum_zero_power
tff(fact_6847_and__numerals_I6_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),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(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).
% and_numerals(6)
tff(fact_6848_and__numerals_I6_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).
% and_numerals(6)
tff(fact_6849_and__numerals_I4_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),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(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).
% and_numerals(4)
tff(fact_6850_and__numerals_I4_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).
% and_numerals(4)
tff(fact_6851_even__take__bit__eq,axiom,
! [Nb: nat,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Nb,A))
<=> ( ( Nb = zero_zero_nat )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).
% even_take_bit_eq
tff(fact_6852_even__take__bit__eq,axiom,
! [Nb: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Nb),A))
<=> ( ( Nb = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).
% even_take_bit_eq
tff(fact_6853_even__take__bit__eq,axiom,
! [Nb: nat,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(Nb),A))
<=> ( ( Nb = zero_zero_nat )
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).
% even_take_bit_eq
tff(fact_6854_real__sqrt__abs,axiom,
! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).
% real_sqrt_abs
tff(fact_6855_and__minus__numerals_I1_J,axiom,
! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).
% and_minus_numerals(1)
tff(fact_6856_and__minus__numerals_I5_J,axiom,
! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).
% and_minus_numerals(5)
tff(fact_6857_sum__zero__power_H,axiom,
! [C: fun_nat_complex,D: fun_nat_complex,A3: set_nat] :
groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_fy(C),D),A3) = $ite(
( aa_set_nat_o(finite_finite_nat,A3)
& aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
zero_zero_complex ) ).
% sum_zero_power'
tff(fact_6858_sum__zero__power_H,axiom,
! [C: fun_nat_rat,D: fun_nat_rat,A3: set_nat] :
groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fz(C),D),A3) = $ite(
( aa_set_nat_o(finite_finite_nat,A3)
& aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)),
zero_zero_rat ) ).
% sum_zero_power'
tff(fact_6859_sum__zero__power_H,axiom,
! [C: fun_nat_real,D: fun_nat_real,A3: set_nat] :
groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(C),D),A3) = $ite(
( aa_set_nat_o(finite_finite_nat,A3)
& aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
zero_zero_real ) ).
% sum_zero_power'
tff(fact_6860_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_6861_take__bit__Suc__0,axiom,
! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_Suc_0
tff(fact_6862_real__sqrt__pow2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).
% real_sqrt_pow2
tff(fact_6863_real__sqrt__pow2__iff,axiom,
! [Xb: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).
% real_sqrt_pow2_iff
tff(fact_6864_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [Xb: real,Y: real,Xaa: 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,Xb),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,Xaa),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,Xb),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,Xaa),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_6865_and__numerals_I7_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),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(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).
% and_numerals(7)
tff(fact_6866_and__numerals_I7_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).
% and_numerals(7)
tff(fact_6867_or__numerals_I7_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).
% or_numerals(7)
tff(fact_6868_or__numerals_I7_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).
% or_numerals(7)
tff(fact_6869_or__numerals_I6_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).
% or_numerals(6)
tff(fact_6870_or__numerals_I6_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).
% or_numerals(6)
tff(fact_6871_or__numerals_I4_J,axiom,
! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).
% or_numerals(4)
tff(fact_6872_or__numerals_I4_J,axiom,
! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).
% or_numerals(4)
tff(fact_6873_take__bit__of__exp,axiom,
! [Mb: nat,Nb: nat] : bit_se1745604003318907178nteger(Mb,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).
% take_bit_of_exp
tff(fact_6874_take__bit__of__exp,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% take_bit_of_exp
tff(fact_6875_take__bit__of__exp,axiom,
! [Mb: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% take_bit_of_exp
tff(fact_6876_take__bit__of__2,axiom,
! [Nb: nat] : bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_6877_take__bit__of__2,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_6878_take__bit__of__2,axiom,
! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_6879_take__bit__add,axiom,
! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).
% take_bit_add
tff(fact_6880_take__bit__add,axiom,
! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).
% take_bit_add
tff(fact_6881_take__bit__tightened,axiom,
! [Nb: nat,A: nat,B: nat,Mb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),B) ) ) ) ).
% take_bit_tightened
tff(fact_6882_take__bit__tightened,axiom,
! [Nb: nat,A: int,B: int,Mb: nat] :
( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_int_int(bit_se2923211474154528505it_int(Mb),A) = aa_int_int(bit_se2923211474154528505it_int(Mb),B) ) ) ) ).
% take_bit_tightened
tff(fact_6883_take__bit__tightened__less__eq__nat,axiom,
! [Mb: nat,Nb: nat,Q3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q3)) ) ).
% take_bit_tightened_less_eq_nat
tff(fact_6884_take__bit__nat__less__eq__self,axiom,
! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).
% take_bit_nat_less_eq_self
tff(fact_6885_real__sqrt__less__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).
% real_sqrt_less_mono
tff(fact_6886_exp__less__cancel,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% exp_less_cancel
tff(fact_6887_real__sqrt__mult,axiom,
! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).
% real_sqrt_mult
tff(fact_6888_exp__times__arg__commute,axiom,
! [A3: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A3)),A3) = aa_complex_complex(times_times_complex(A3),aa_complex_complex(exp_complex,A3)) ).
% exp_times_arg_commute
tff(fact_6889_exp__times__arg__commute,axiom,
! [A3: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A3)),A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A3),aa_real_real(exp_real,A3)) ).
% exp_times_arg_commute
tff(fact_6890_take__bit__mult,axiom,
! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).
% take_bit_mult
tff(fact_6891_bit_Ocomplement__unique,axiom,
! [A: code_integer,Xb: code_integer,Y: code_integer] :
( ( bit_se3949692690581998587nteger(A,Xb) = zero_z3403309356797280102nteger )
=> ( ( bit_se1080825931792720795nteger(A,Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
=> ( ( bit_se3949692690581998587nteger(A,Y) = zero_z3403309356797280102nteger )
=> ( ( bit_se1080825931792720795nteger(A,Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
=> ( Xb = Y ) ) ) ) ) ).
% bit.complement_unique
tff(fact_6892_bit_Ocomplement__unique,axiom,
! [A: int,Xb: int,Y: int] :
( ( aa_int_int(bit_se725231765392027082nd_int(A),Xb) = zero_zero_int )
=> ( ( aa_int_int(bit_se1409905431419307370or_int(A),Xb) = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( aa_int_int(bit_se725231765392027082nd_int(A),Y) = zero_zero_int )
=> ( ( aa_int_int(bit_se1409905431419307370or_int(A),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( Xb = Y ) ) ) ) ) ).
% bit.complement_unique
tff(fact_6893_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_6894_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( aa_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_6895_real__sqrt__gt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).
% real_sqrt_gt_zero
tff(fact_6896_exp__total,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ? [X4: real] : aa_real_real(exp_real,X4) = Y ) ).
% exp_total
tff(fact_6897_exp__gt__zero,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).
% exp_gt_zero
tff(fact_6898_not__exp__less__zero,axiom,
! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).
% not_exp_less_zero
tff(fact_6899_sum__cong__Suc,axiom,
! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
( ~ aa_set_nat_o(member_nat(zero_zero_nat),A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(aa_nat_nat(suc,X4)),A3)
=> ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ) ).
% sum_cong_Suc
tff(fact_6900_sum__cong__Suc,axiom,
! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
( ~ aa_set_nat_o(member_nat(zero_zero_nat),A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(aa_nat_nat(suc,X4)),A3)
=> ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
=> ( groups6591440286371151544t_real(F,A3) = groups6591440286371151544t_real(G,A3) ) ) ) ).
% sum_cong_Suc
tff(fact_6901_take__bit__tightened__less__eq__int,axiom,
! [Mb: nat,Nb: nat,K: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).
% take_bit_tightened_less_eq_int
tff(fact_6902_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [Nb: nat,A: int,B: int] :
( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
<=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6903_take__bit__int__greater__self__iff,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% take_bit_int_greater_self_iff
tff(fact_6904_not__take__bit__negative,axiom,
! [Nb: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),zero_zero_int) ).
% not_take_bit_negative
tff(fact_6905_signed__take__bit__take__bit,axiom,
! [Mb: nat,Nb: nat,A: int] :
aa_int_int(bit_ri631733984087533419it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Mb)),A) ).
% signed_take_bit_take_bit
tff(fact_6906_mult__exp__exp,axiom,
! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).
% mult_exp_exp
tff(fact_6907_mult__exp__exp,axiom,
! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).
% mult_exp_exp
tff(fact_6908_exp__add__commuting,axiom,
! [Xb: complex,Y: complex] :
( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
=> ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ) ) ).
% exp_add_commuting
tff(fact_6909_exp__add__commuting,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
=> ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ) ).
% exp_add_commuting
tff(fact_6910_take__bit__unset__bit__eq,axiom,
! [Nb: nat,Mb: nat,A: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).
% take_bit_unset_bit_eq
tff(fact_6911_take__bit__unset__bit__eq,axiom,
! [Nb: nat,Mb: nat,A: int] :
aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se4203085406695923979it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se4203085406695923979it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).
% take_bit_unset_bit_eq
tff(fact_6912_take__bit__set__bit__eq,axiom,
! [Nb: nat,Mb: nat,A: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).
% take_bit_set_bit_eq
tff(fact_6913_take__bit__set__bit__eq,axiom,
! [Nb: nat,Mb: nat,A: int] :
aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se7879613467334960850it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se7879613467334960850it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).
% take_bit_set_bit_eq
tff(fact_6914_take__bit__flip__bit__eq,axiom,
! [Nb: nat,Mb: nat,A: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).
% take_bit_flip_bit_eq
tff(fact_6915_take__bit__flip__bit__eq,axiom,
! [Nb: nat,Mb: nat,A: int] :
aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).
% take_bit_flip_bit_eq
tff(fact_6916_sum__subtractf__nat,axiom,
! [A3: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
( ! [X4: product_prod_nat_nat] :
( member8440522571783428010at_nat(X4,A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X4)),aa_Pro4243173770044397973at_nat(F,X4)) )
=> ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_gb(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A3)),groups977919841031483927at_nat(G,A3)) ) ) ).
% sum_subtractf_nat
tff(fact_6917_sum__subtractf__nat,axiom,
! [A3: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X4)),aa_complex_nat(F,X4)) )
=> ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gc(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ).
% sum_subtractf_nat
tff(fact_6918_sum__subtractf__nat,axiom,
! [A3: set_real,G: fun_real_nat,F: fun_real_nat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4)) )
=> ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gd(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(G,A3)) ) ) ).
% sum_subtractf_nat
tff(fact_6919_sum__subtractf__nat,axiom,
! [A3: set_int,G: fun_int_nat,F: fun_int_nat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4)) )
=> ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ge(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ).
% sum_subtractf_nat
tff(fact_6920_sum__subtractf__nat,axiom,
! [A3: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4)) )
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gf(G),F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ).
% sum_subtractf_nat
tff(fact_6921_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gg(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.shift_bounds_cl_Suc_ivl
tff(fact_6922_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups6591440286371151544t_real(aTP_Lamm_gh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.shift_bounds_cl_Suc_ivl
tff(fact_6923_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: fun_nat_nat,Mb: nat,K: nat,Nb: 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,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gi(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.shift_bounds_cl_nat_ivl
tff(fact_6924_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: fun_nat_real,Mb: nat,K: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gj(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.shift_bounds_cl_nat_ivl
tff(fact_6925_pow_Osimps_I1_J,axiom,
! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).
% pow.simps(1)
tff(fact_6926_sum__eq__Suc0__iff,axiom,
! [A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( groups4541462559716669496nt_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X: int] :
( aa_set_int_o(member_int(X),A3)
& ( aa_int_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
& ! [Xa2: int] :
( aa_set_int_o(member_int(Xa2),A3)
=> ( ( X != Xa2 )
=> ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_6927_sum__eq__Suc0__iff,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ( groups5693394587270226106ex_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
& ( aa_complex_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
& ! [Xa2: complex] :
( aa_set_complex_o(member_complex(Xa2),A3)
=> ( ( X != Xa2 )
=> ( aa_complex_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_6928_sum__eq__Suc0__iff,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
& ( aa_nat_nat(F,X) = aa_nat_nat(suc,zero_zero_nat) )
& ! [Xa2: nat] :
( aa_set_nat_o(member_nat(Xa2),A3)
=> ( ( X != Xa2 )
=> ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_6929_sum__SucD,axiom,
! [F: fun_nat_nat,A3: set_nat,Nb: nat] :
( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,Nb) )
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ).
% sum_SucD
tff(fact_6930_sum__eq__1__iff,axiom,
! [A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( groups4541462559716669496nt_nat(F,A3) = one_one_nat )
<=> ? [X: int] :
( aa_set_int_o(member_int(X),A3)
& ( aa_int_nat(F,X) = one_one_nat )
& ! [Xa2: int] :
( aa_set_int_o(member_int(Xa2),A3)
=> ( ( X != Xa2 )
=> ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_6931_sum__eq__1__iff,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ( groups5693394587270226106ex_nat(F,A3) = one_one_nat )
<=> ? [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
& ( aa_complex_nat(F,X) = one_one_nat )
& ! [Xa2: complex] :
( aa_set_complex_o(member_complex(Xa2),A3)
=> ( ( X != Xa2 )
=> ( aa_complex_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_6932_sum__eq__1__iff,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = one_one_nat )
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
& ( aa_nat_nat(F,X) = one_one_nat )
& ! [Xa2: nat] :
( aa_set_nat_o(member_nat(Xa2),A3)
=> ( ( X != Xa2 )
=> ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_6933_exp__gt__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb)) ) ).
% exp_gt_one
tff(fact_6934_take__bit__signed__take__bit,axiom,
! [Mb: nat,Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Mb),A) ) ) ).
% take_bit_signed_take_bit
tff(fact_6935_le__real__sqrt__sumsq,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).
% le_real_sqrt_sumsq
tff(fact_6936_and__less__eq,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K) ) ).
% and_less_eq
tff(fact_6937_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).
% AND_upper1''
tff(fact_6938_AND__upper2_H_H,axiom,
! [Y: int,Z: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),Z) ) ) ).
% AND_upper2''
tff(fact_6939_exp__minus__inverse,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) = one_one_real ).
% exp_minus_inverse
tff(fact_6940_exp__minus__inverse,axiom,
! [Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb))) = one_one_complex ).
% exp_minus_inverse
tff(fact_6941_sum__power__add,axiom,
! [Xb: complex,Mb: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_gk(Xb),Mb),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),groups2073611262835488442omplex(power_power_complex(Xb),I5)) ).
% sum_power_add
tff(fact_6942_sum__power__add,axiom,
! [Xb: rat,Mb: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_gl(Xb),Mb),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),I5)) ).
% sum_power_add
tff(fact_6943_sum__power__add,axiom,
! [Xb: int,Mb: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_gm(Xb),Mb),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),I5)) ).
% sum_power_add
tff(fact_6944_sum__power__add,axiom,
! [Xb: real,Mb: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gn(Xb),Mb),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),I5)) ).
% sum_power_add
tff(fact_6945_sum_OatLeastAtMost__rev,axiom,
! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_go(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).
% sum.atLeastAtMost_rev
tff(fact_6946_sum_OatLeastAtMost__rev,axiom,
! [G: fun_nat_real,Nb: nat,Mb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gp(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).
% sum.atLeastAtMost_rev
tff(fact_6947_sum__roots__unity,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
=> ( groups7754918857620584856omplex(aTP_Lamm_ag,collect_complex(aTP_Lamm_bp(Nb))) = zero_zero_complex ) ) ).
% sum_roots_unity
tff(fact_6948_sum__nth__roots,axiom,
! [Nb: nat,C: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
=> ( groups7754918857620584856omplex(aTP_Lamm_ag,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_bn(Nb),C))) = zero_zero_complex ) ) ).
% sum_nth_roots
tff(fact_6949_even__and__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).
% even_and_iff
tff(fact_6950_even__and__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(A),B))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).
% even_and_iff
tff(fact_6951_even__and__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
| aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).
% even_and_iff
tff(fact_6952_sqrt2__less__2,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% sqrt2_less_2
tff(fact_6953_even__or__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1080825931792720795nteger(A,B))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).
% even_or_iff
tff(fact_6954_even__or__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(A),B))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).
% even_or_iff
tff(fact_6955_even__or__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).
% even_or_iff
tff(fact_6956_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_6957_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_6958_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_6959_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_6960_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_6961_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6962_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6963_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = 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),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6964_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6965_even__and__iff__int,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(K),L))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).
% even_and_iff_int
tff(fact_6966_sum_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6967_sum_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6968_sum_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6969_sum_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6970_sum_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6971_sum_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6972_sum_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6973_sum_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6974_ln__ge__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xb) ) ) ).
% ln_ge_iff
tff(fact_6975_sum_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_gq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6976_sum_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_gr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6977_sum_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( 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(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gg(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6978_sum_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),groups6591440286371151544t_real(aTP_Lamm_gh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6979_sum__Suc__diff,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups2906978787729119204at_rat(aTP_Lamm_gs(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Mb)) ) ) ).
% sum_Suc_diff
tff(fact_6980_sum__Suc__diff,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups3539618377306564664at_int(aTP_Lamm_gt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).
% sum_Suc_diff
tff(fact_6981_sum__Suc__diff,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups6591440286371151544t_real(aTP_Lamm_gu(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).
% sum_Suc_diff
tff(fact_6982_take__bit__Suc__bit0,axiom,
! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),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,Nb),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_6983_take__bit__Suc__bit0,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_6984_take__bit__eq__mod,axiom,
! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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))),Nb)) ).
% take_bit_eq_mod
tff(fact_6985_take__bit__eq__mod,axiom,
! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% take_bit_eq_mod
tff(fact_6986_and__one__eq,axiom,
! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_6987_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_6988_one__and__eq,axiom,
! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_6989_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_6990_take__bit__nat__eq__self,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).
% take_bit_nat_eq_self
tff(fact_6991_take__bit__nat__less__exp,axiom,
! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% take_bit_nat_less_exp
tff(fact_6992_take__bit__nat__eq__self__iff,axiom,
! [Nb: nat,Mb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).
% take_bit_nat_eq_self_iff
tff(fact_6993_real__less__rsqrt,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y)) ) ).
% real_less_rsqrt
tff(fact_6994_sqrt__le__D,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_6995_real__le__rsqrt,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y)) ) ).
% real_le_rsqrt
tff(fact_6996_take__bit__nat__def,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% take_bit_nat_def
tff(fact_6997_exp__le,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).
% exp_le
tff(fact_6998_sum_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_6999_sum_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_7000_sum_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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(Mb),Nb))),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,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_7001_sum_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_7002_take__bit__int__less__exp,axiom,
! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% take_bit_int_less_exp
tff(fact_7003_take__bit__int__def,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(modulo_modulo_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% take_bit_int_def
tff(fact_7004_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_7005_tanh__altdef,axiom,
! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).
% tanh_altdef
tff(fact_7006_tanh__altdef,axiom,
! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).
% tanh_altdef
tff(fact_7007_take__bit__eq__0__iff,axiom,
! [Nb: nat,A: code_integer] :
( ( bit_se1745604003318907178nteger(Nb,A) = zero_z3403309356797280102nteger )
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)),A) ) ).
% take_bit_eq_0_iff
tff(fact_7008_take__bit__eq__0__iff,axiom,
! [Nb: nat,A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = zero_zero_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(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))),Nb)),A) ) ).
% take_bit_eq_0_iff
tff(fact_7009_take__bit__eq__0__iff,axiom,
! [Nb: nat,A: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = zero_zero_int )
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),A) ) ).
% take_bit_eq_0_iff
tff(fact_7010_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_7011_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_7012_take__bit__nat__less__self__iff,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
<=> aa_nat_o(aa_nat_fun_nat_o(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))),Nb)),Mb) ) ).
% take_bit_nat_less_self_iff
tff(fact_7013_real__le__lsqrt,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).
% real_le_lsqrt
tff(fact_7014_real__sqrt__unique,axiom,
! [Y: real,Xb: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_real(sqrt,Xb) = Y ) ) ) ).
% real_sqrt_unique
tff(fact_7015_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).
% lemma_real_divide_sqrt_less
tff(fact_7016_real__sqrt__sum__squares__eq__cancel,axiom,
! [Xb: 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,Xb),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))))) = Xb )
=> ( Y = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel
tff(fact_7017_real__sqrt__sum__squares__eq__cancel2,axiom,
! [Xb: 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,Xb),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 )
=> ( Xb = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel2
tff(fact_7018_real__sqrt__sum__squares__ge1,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7019_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7020_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,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_7021_exp__half__le2,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% exp_half_le2
tff(fact_7022_sqrt__ge__absD,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).
% sqrt_ge_absD
tff(fact_7023_take__bit__Suc__minus__bit0,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),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_7024_mask__Suc__exp,axiom,
! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).
% mask_Suc_exp
tff(fact_7025_mask__Suc__exp,axiom,
! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).
% mask_Suc_exp
tff(fact_7026_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).
% take_bit_int_greater_eq_self_iff
tff(fact_7027_take__bit__int__less__self__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),K)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).
% take_bit_int_less_self_iff
tff(fact_7028_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_7029_exp__double,axiom,
! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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_7030_sum__natinterval__diff,axiom,
! [F: fun_nat_complex,Mb: nat,Nb: nat] :
groups2073611262835488442omplex(aTP_Lamm_gv(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).
% sum_natinterval_diff
tff(fact_7031_sum__natinterval__diff,axiom,
! [F: fun_nat_rat,Mb: nat,Nb: nat] :
groups2906978787729119204at_rat(aTP_Lamm_gw(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).
% sum_natinterval_diff
tff(fact_7032_sum__natinterval__diff,axiom,
! [F: fun_nat_int,Mb: nat,Nb: nat] :
groups3539618377306564664at_int(aTP_Lamm_gx(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).
% sum_natinterval_diff
tff(fact_7033_sum__natinterval__diff,axiom,
! [F: fun_nat_real,Mb: nat,Nb: nat] :
groups6591440286371151544t_real(aTP_Lamm_gy(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).
% sum_natinterval_diff
tff(fact_7034_sum__telescope_H_H,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups2906978787729119204at_rat(aTP_Lamm_gz(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb)) ) ) ).
% sum_telescope''
tff(fact_7035_sum__telescope_H_H,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups3539618377306564664at_int(aTP_Lamm_ha(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).
% sum_telescope''
tff(fact_7036_sum__telescope_H_H,axiom,
! [Mb: nat,Nb: nat,F: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups6591440286371151544t_real(aTP_Lamm_hb(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).
% sum_telescope''
tff(fact_7037_real__less__lsqrt,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).
% real_less_lsqrt
tff(fact_7038_sqrt__sum__squares__le__sum,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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,Xb),Y)) ) ) ).
% sqrt_sum_squares_le_sum
tff(fact_7039_one__or__eq,axiom,
! [A: code_integer] : bit_se1080825931792720795nteger(one_one_Code_integer,A) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_7040_one__or__eq,axiom,
! [A: int] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_7041_one__or__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_7042_or__one__eq,axiom,
! [A: code_integer] : bit_se1080825931792720795nteger(A,one_one_Code_integer) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_7043_or__one__eq,axiom,
! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_7044_or__one__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_7045_sqrt__even__pow2,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% sqrt_even_pow2
tff(fact_7046_mask__Suc__double,axiom,
! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_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,Nb))) ).
% mask_Suc_double
tff(fact_7047_mask__Suc__double,axiom,
! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(bit_se1409905431419307370or_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,Nb))) ).
% mask_Suc_double
tff(fact_7048_real__sqrt__ge__abs1,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7049_real__sqrt__ge__abs2,axiom,
! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7050_sqrt__sum__squares__le__sum__abs,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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,Xb)),aa_real_real(abs_abs_real,Y))) ).
% sqrt_sum_squares_le_sum_abs
tff(fact_7051_ln__sqrt,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).
% ln_sqrt
tff(fact_7052_OR__upper,axiom,
! [Xb: int,Nb: nat,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).
% OR_upper
tff(fact_7053_or__int__rec,axiom,
! [K: int,L: int] :
aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
| ~ aa_int_o(aa_int_fun_int_o(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(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% or_int_rec
tff(fact_7054_take__bit__int__eq__self__iff,axiom,
! [Nb: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).
% take_bit_int_eq_self_iff
tff(fact_7055_take__bit__int__eq__self,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K ) ) ) ).
% take_bit_int_eq_self
tff(fact_7056_and__int__rec,axiom,
! [K: int,L: int] :
aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% and_int_rec
tff(fact_7057_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_7058_take__bit__incr__eq,axiom,
! [Nb: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),one_one_int) )
=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),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(Nb),K)) ) ) ).
% take_bit_incr_eq
tff(fact_7059_take__bit__eq__mask__iff__exp__dvd,axiom,
! [Nb: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).
% take_bit_eq_mask_iff_exp_dvd
tff(fact_7060_mask__eq__sum__exp,axiom,
! [Nb: 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))),Nb)),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_o(aTP_Lamm_bj,Nb))) ).
% mask_eq_sum_exp
tff(fact_7061_mask__eq__sum__exp,axiom,
! [Nb: 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))),Nb)),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_o(aTP_Lamm_bj,Nb))) ).
% mask_eq_sum_exp
tff(fact_7062_sum__gp__multiplied,axiom,
! [Mb: nat,Nb: nat,Xb: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ) ) ).
% sum_gp_multiplied
tff(fact_7063_sum__gp__multiplied,axiom,
! [Mb: nat,Nb: nat,Xb: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).
% sum_gp_multiplied
tff(fact_7064_sum__gp__multiplied,axiom,
! [Mb: nat,Nb: nat,Xb: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( 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),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ) ) ).
% sum_gp_multiplied
tff(fact_7065_sum__gp__multiplied,axiom,
! [Mb: nat,Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).
% sum_gp_multiplied
tff(fact_7066_sum_Oin__pairs,axiom,
! [G: fun_nat_rat,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_hc(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.in_pairs
tff(fact_7067_sum_Oin__pairs,axiom,
! [G: fun_nat_int,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_hd(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.in_pairs
tff(fact_7068_sum_Oin__pairs,axiom,
! [G: fun_nat_nat,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_he(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.in_pairs
tff(fact_7069_sum_Oin__pairs,axiom,
! [G: fun_nat_real,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_hf(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% sum.in_pairs
tff(fact_7070_take__bit__Suc__bit1,axiom,
! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_7071_take__bit__Suc__bit1,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),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_7072_take__bit__Suc__minus__1__eq,axiom,
! [Nb: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),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,Nb)),one_one_Code_integer) ).
% take_bit_Suc_minus_1_eq
tff(fact_7073_take__bit__Suc__minus__1__eq,axiom,
! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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,Nb))),one_one_int) ).
% take_bit_Suc_minus_1_eq
tff(fact_7074_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_7075_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_7076_take__bit__Suc,axiom,
! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_7077_take__bit__Suc,axiom,
! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),aa_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(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_7078_arsinh__real__aux,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).
% arsinh_real_aux
tff(fact_7079_real__sqrt__power__even,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% real_sqrt_power_even
tff(fact_7080_exp__bound,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).
% exp_bound
tff(fact_7081_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(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,Xb),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,Xaa),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_7082_arith__geo__mean__sqrt,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).
% arith_geo_mean_sqrt
tff(fact_7083_take__bit__int__less__eq,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb))) ) ) ).
% take_bit_int_less_eq
tff(fact_7084_take__bit__int__greater__eq,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).
% take_bit_int_greater_eq
tff(fact_7085_signed__take__bit__eq__take__bit__shift,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% signed_take_bit_eq_take_bit_shift
tff(fact_7086_mask__eq__sum__exp__nat,axiom,
! [Nb: 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))),Nb)),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_o(aTP_Lamm_bj,Nb))) ).
% mask_eq_sum_exp_nat
tff(fact_7087_stable__imp__take__bit__eq,axiom,
! [A: code_integer,Nb: nat] :
( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
=> ( bit_se1745604003318907178nteger(Nb,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_7088_stable__imp__take__bit__eq,axiom,
! [A: nat,Nb: 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(bit_se2925701944663578781it_nat,Nb),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),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))),Nb)),one_one_nat)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_7089_stable__imp__take__bit__eq,axiom,
! [A: int,Nb: nat] :
( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_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))),Nb)),one_one_int)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_7090_gauss__sum__nat,axiom,
! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum_nat
tff(fact_7091_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_7092_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_7093_real__exp__bound__lemma,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).
% real_exp_bound_lemma
tff(fact_7094_cos__x__y__le__one,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7095_real__sqrt__sum__squares__less,axiom,
! [Xb: real,U: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7096_arcosh__real__def,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
=> ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).
% arcosh_real_def
tff(fact_7097_take__bit__minus__small__eq,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),K) ) ) ) ).
% take_bit_minus_small_eq
tff(fact_7098_arith__series__nat,axiom,
! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hg(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),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,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% arith_series_nat
tff(fact_7099_Sum__Icc__nat,axiom,
! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = 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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Sum_Icc_nat
tff(fact_7100_sqrt__sum__squares__half__less,axiom,
! [Xb: real,U: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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_7101_exp__lower__Taylor__quadratic,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).
% exp_lower_Taylor_quadratic
tff(fact_7102_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,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_numeral_minus_bit1
tff(fact_7103_arsinh__real__def,axiom,
! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).
% arsinh_real_def
tff(fact_7104_infinite__int__iff__unbounded,axiom,
! [S3: set_int] :
( ~ aa_set_int_o(finite_finite_int,S3)
<=> ! [M4: int] :
? [N3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,M4),aa_int_int(abs_abs_int,N3))
& aa_set_int_o(member_int(N3),S3) ) ) ).
% infinite_int_iff_unbounded
tff(fact_7105_take__bit__Suc__minus__bit1,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_Suc_minus_bit1
tff(fact_7106_and__int_Oelims,axiom,
! [Xb: int,Xaa: int,Y: int] :
( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
=> ( Y = $ite(
( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),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_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).
% and_int.elims
tff(fact_7107_and__int_Osimps,axiom,
! [K: int,L: int] :
aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).
% and_int.simps
tff(fact_7108_pred__numeral__inc,axiom,
! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).
% pred_numeral_inc
tff(fact_7109_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_7110_and__nat__numerals_I3_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
% and_nat_numerals(3)
tff(fact_7111_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_7112_or__nat__numerals_I4_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).
% or_nat_numerals(4)
tff(fact_7113_sum_Oinsert,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7114_sum_Oinsert,axiom,
! [A3: set_real,Xb: real,G: fun_real_real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ~ aa_set_real_o(member_real(Xb),A3)
=> ( groups8097168146408367636l_real(G,insert_real(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xb)),groups8097168146408367636l_real(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7115_sum_Oinsert,axiom,
! [A3: set_int,Xb: int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ~ aa_set_int_o(member_int(Xb),A3)
=> ( groups8778361861064173332t_real(G,insert_int(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xb)),groups8778361861064173332t_real(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7116_sum_Oinsert,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ~ aa_set_complex_o(member_complex(Xb),A3)
=> ( groups5808333547571424918x_real(G,insert_complex(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7117_sum_Oinsert,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7118_sum_Oinsert,axiom,
! [A3: set_real,Xb: real,G: fun_real_rat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ~ aa_set_real_o(member_real(Xb),A3)
=> ( groups1300246762558778688al_rat(G,insert_real(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xb)),groups1300246762558778688al_rat(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7119_sum_Oinsert,axiom,
! [A3: set_nat,Xb: nat,G: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ~ aa_set_nat_o(member_nat(Xb),A3)
=> ( groups2906978787729119204at_rat(G,insert_nat(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xb)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7120_sum_Oinsert,axiom,
! [A3: set_int,Xb: int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ~ aa_set_int_o(member_int(Xb),A3)
=> ( groups3906332499630173760nt_rat(G,insert_int(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7121_sum_Oinsert,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ~ aa_set_complex_o(member_complex(Xb),A3)
=> ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7122_sum_Oinsert,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,A3)) ) ) ) ).
% sum.insert
tff(fact_7123_or__nat__numerals_I3_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).
% or_nat_numerals(3)
tff(fact_7124_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_7125_and__nat__numerals_I4_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% and_nat_numerals(4)
tff(fact_7126_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_7127_set__replicate,axiom,
! [Nb: nat,Xb: vEBT_VEBT] :
( ( Nb != zero_zero_nat )
=> ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ) ) ).
% set_replicate
tff(fact_7128_set__replicate,axiom,
! [Nb: nat,Xb: nat] :
( ( Nb != zero_zero_nat )
=> ( set_nat2(replicate_nat(Nb,Xb)) = insert_nat(Xb,bot_bot_set_nat) ) ) ).
% set_replicate
tff(fact_7129_set__replicate,axiom,
! [Nb: nat,Xb: int] :
( ( Nb != zero_zero_nat )
=> ( set_int2(replicate_int(Nb,Xb)) = insert_int(Xb,bot_bot_set_int) ) ) ).
% set_replicate
tff(fact_7130_set__replicate,axiom,
! [Nb: nat,Xb: real] :
( ( Nb != zero_zero_nat )
=> ( set_real2(replicate_real(Nb,Xb)) = insert_real(Xb,bot_bot_set_real) ) ) ).
% set_replicate
tff(fact_7131_add__neg__numeral__special_I6_J,axiom,
! [Mb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb))) ).
% add_neg_numeral_special(6)
tff(fact_7132_add__neg__numeral__special_I6_J,axiom,
! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb))) ).
% add_neg_numeral_special(6)
tff(fact_7133_add__neg__numeral__special_I6_J,axiom,
! [Mb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Mb))) ).
% add_neg_numeral_special(6)
tff(fact_7134_add__neg__numeral__special_I6_J,axiom,
! [Mb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb))) ).
% add_neg_numeral_special(6)
tff(fact_7135_add__neg__numeral__special_I6_J,axiom,
! [Mb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Mb))) ).
% add_neg_numeral_special(6)
tff(fact_7136_add__neg__numeral__special_I5_J,axiom,
! [Nb: 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,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).
% add_neg_numeral_special(5)
tff(fact_7137_add__neg__numeral__special_I5_J,axiom,
! [Nb: 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,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).
% add_neg_numeral_special(5)
tff(fact_7138_add__neg__numeral__special_I5_J,axiom,
! [Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).
% add_neg_numeral_special(5)
tff(fact_7139_add__neg__numeral__special_I5_J,axiom,
! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).
% add_neg_numeral_special(5)
tff(fact_7140_add__neg__numeral__special_I5_J,axiom,
! [Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).
% add_neg_numeral_special(5)
tff(fact_7141_diff__numeral__special_I5_J,axiom,
! [Nb: 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,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).
% diff_numeral_special(5)
tff(fact_7142_diff__numeral__special_I5_J,axiom,
! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).
% diff_numeral_special(5)
tff(fact_7143_diff__numeral__special_I5_J,axiom,
! [Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).
% diff_numeral_special(5)
tff(fact_7144_diff__numeral__special_I5_J,axiom,
! [Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).
% diff_numeral_special(5)
tff(fact_7145_diff__numeral__special_I5_J,axiom,
! [Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).
% diff_numeral_special(5)
tff(fact_7146_diff__numeral__special_I6_J,axiom,
! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb)) ).
% diff_numeral_special(6)
tff(fact_7147_diff__numeral__special_I6_J,axiom,
! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb)) ).
% diff_numeral_special(6)
tff(fact_7148_diff__numeral__special_I6_J,axiom,
! [Mb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,Mb)) ).
% diff_numeral_special(6)
tff(fact_7149_diff__numeral__special_I6_J,axiom,
! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb)) ).
% diff_numeral_special(6)
tff(fact_7150_diff__numeral__special_I6_J,axiom,
! [Mb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(inc,Mb)) ).
% diff_numeral_special(6)
tff(fact_7151_and__Suc__0__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% and_Suc_0_eq
tff(fact_7152_Suc__0__and__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Suc_0_and_eq
tff(fact_7153_num__induct,axiom,
! [P: fun_num_o,Xb: num] :
( aa_num_o(P,one)
=> ( ! [X4: num] :
( aa_num_o(P,X4)
=> aa_num_o(P,aa_num_num(inc,X4)) )
=> aa_num_o(P,Xb) ) ) ).
% num_induct
tff(fact_7154_add__inc,axiom,
! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y)) ).
% add_inc
tff(fact_7155_inc_Osimps_I1_J,axiom,
aa_num_num(inc,one) = aa_num_num(bit0,one) ).
% inc.simps(1)
tff(fact_7156_inc_Osimps_I2_J,axiom,
! [Xb: num] : aa_num_num(inc,aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).
% inc.simps(2)
tff(fact_7157_inc_Osimps_I3_J,axiom,
! [Xb: num] : aa_num_num(inc,aa_num_num(bit1,Xb)) = aa_num_num(bit0,aa_num_num(inc,Xb)) ).
% inc.simps(3)
tff(fact_7158_add__One,axiom,
! [Xb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),one) = aa_num_num(inc,Xb) ).
% add_One
tff(fact_7159_inc__BitM__eq,axiom,
! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).
% inc_BitM_eq
tff(fact_7160_BitM__inc__eq,axiom,
! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).
% BitM_inc_eq
tff(fact_7161_finite__ranking__induct,axiom,
! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
=> ( ! [X4: vEBT_VEBT,S4: set_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
=> ( ! [Y4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S4)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y4)),aa_VEBT_VEBT_rat(F,X4)) )
=> ( aa_set_VEBT_VEBT_o(P,S4)
=> aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X4,S4)) ) ) )
=> aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7162_finite__ranking__induct,axiom,
! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( aa_set_complex_o(P,bot_bot_set_complex)
=> ( ! [X4: complex,S4: set_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S4)
=> ( ! [Y4: complex] :
( aa_set_complex_o(member_complex(Y4),S4)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X4)) )
=> ( aa_set_complex_o(P,S4)
=> aa_set_complex_o(P,insert_complex(X4,S4)) ) ) )
=> aa_set_complex_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7163_finite__ranking__induct,axiom,
! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X4: nat,S4: set_nat] :
( aa_set_nat_o(finite_finite_nat,S4)
=> ( ! [Y4: nat] :
( aa_set_nat_o(member_nat(Y4),S4)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X4)) )
=> ( aa_set_nat_o(P,S4)
=> aa_set_nat_o(P,insert_nat(X4,S4)) ) ) )
=> aa_set_nat_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7164_finite__ranking__induct,axiom,
! [S3: set_int,P: fun_set_int_o,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,S3)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X4: int,S4: set_int] :
( aa_set_int_o(finite_finite_int,S4)
=> ( ! [Y4: int] :
( aa_set_int_o(member_int(Y4),S4)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X4)) )
=> ( aa_set_int_o(P,S4)
=> aa_set_int_o(P,insert_int(X4,S4)) ) ) )
=> aa_set_int_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7165_finite__ranking__induct,axiom,
! [S3: set_real,P: fun_set_real_o,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,S3)
=> ( aa_set_real_o(P,bot_bot_set_real)
=> ( ! [X4: real,S4: set_real] :
( aa_set_real_o(finite_finite_real,S4)
=> ( ! [Y4: real] :
( aa_set_real_o(member_real(Y4),S4)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X4)) )
=> ( aa_set_real_o(P,S4)
=> aa_set_real_o(P,insert_real(X4,S4)) ) ) )
=> aa_set_real_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7166_finite__ranking__induct,axiom,
! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_num] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
=> ( ! [X4: vEBT_VEBT,S4: set_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
=> ( ! [Y4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S4)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y4)),aa_VEBT_VEBT_num(F,X4)) )
=> ( aa_set_VEBT_VEBT_o(P,S4)
=> aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X4,S4)) ) ) )
=> aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7167_finite__ranking__induct,axiom,
! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( aa_set_complex_o(P,bot_bot_set_complex)
=> ( ! [X4: complex,S4: set_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S4)
=> ( ! [Y4: complex] :
( aa_set_complex_o(member_complex(Y4),S4)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X4)) )
=> ( aa_set_complex_o(P,S4)
=> aa_set_complex_o(P,insert_complex(X4,S4)) ) ) )
=> aa_set_complex_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7168_finite__ranking__induct,axiom,
! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X4: nat,S4: set_nat] :
( aa_set_nat_o(finite_finite_nat,S4)
=> ( ! [Y4: nat] :
( aa_set_nat_o(member_nat(Y4),S4)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y4)),aa_nat_num(F,X4)) )
=> ( aa_set_nat_o(P,S4)
=> aa_set_nat_o(P,insert_nat(X4,S4)) ) ) )
=> aa_set_nat_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7169_finite__ranking__induct,axiom,
! [S3: set_int,P: fun_set_int_o,F: fun_int_num] :
( aa_set_int_o(finite_finite_int,S3)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X4: int,S4: set_int] :
( aa_set_int_o(finite_finite_int,S4)
=> ( ! [Y4: int] :
( aa_set_int_o(member_int(Y4),S4)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y4)),aa_int_num(F,X4)) )
=> ( aa_set_int_o(P,S4)
=> aa_set_int_o(P,insert_int(X4,S4)) ) ) )
=> aa_set_int_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7170_finite__ranking__induct,axiom,
! [S3: set_real,P: fun_set_real_o,F: fun_real_num] :
( aa_set_real_o(finite_finite_real,S3)
=> ( aa_set_real_o(P,bot_bot_set_real)
=> ( ! [X4: real,S4: set_real] :
( aa_set_real_o(finite_finite_real,S4)
=> ( ! [Y4: real] :
( aa_set_real_o(member_real(Y4),S4)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X4)) )
=> ( aa_set_real_o(P,S4)
=> aa_set_real_o(P,insert_real(X4,S4)) ) ) )
=> aa_set_real_o(P,S3) ) ) ) ).
% finite_ranking_induct
tff(fact_7171_finite__linorder__min__induct,axiom,
! [A3: set_real,P: fun_set_real_o] :
( aa_set_real_o(finite_finite_real,A3)
=> ( aa_set_real_o(P,bot_bot_set_real)
=> ( ! [B4: real,A6: set_real] :
( aa_set_real_o(finite_finite_real,A6)
=> ( ! [X3: real] :
( aa_set_real_o(member_real(X3),A6)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B4),X3) )
=> ( aa_set_real_o(P,A6)
=> aa_set_real_o(P,insert_real(B4,A6)) ) ) )
=> aa_set_real_o(P,A3) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7172_finite__linorder__min__induct,axiom,
! [A3: set_rat,P: fun_set_rat_o] :
( aa_set_rat_o(finite_finite_rat,A3)
=> ( aa_set_rat_o(P,bot_bot_set_rat)
=> ( ! [B4: rat,A6: set_rat] :
( aa_set_rat_o(finite_finite_rat,A6)
=> ( ! [X3: rat] :
( aa_set_rat_o(member_rat(X3),A6)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),X3) )
=> ( aa_set_rat_o(P,A6)
=> aa_set_rat_o(P,insert_rat(B4,A6)) ) ) )
=> aa_set_rat_o(P,A3) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7173_finite__linorder__min__induct,axiom,
! [A3: set_num,P: fun_set_num_o] :
( aa_set_num_o(finite_finite_num,A3)
=> ( aa_set_num_o(P,bot_bot_set_num)
=> ( ! [B4: num,A6: set_num] :
( aa_set_num_o(finite_finite_num,A6)
=> ( ! [X3: num] :
( aa_set_num_o(member_num(X3),A6)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,B4),X3) )
=> ( aa_set_num_o(P,A6)
=> aa_set_num_o(P,insert_num(B4,A6)) ) ) )
=> aa_set_num_o(P,A3) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7174_finite__linorder__min__induct,axiom,
! [A3: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [B4: nat,A6: set_nat] :
( aa_set_nat_o(finite_finite_nat,A6)
=> ( ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),A6)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B4),X3) )
=> ( aa_set_nat_o(P,A6)
=> aa_set_nat_o(P,insert_nat(B4,A6)) ) ) )
=> aa_set_nat_o(P,A3) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7175_finite__linorder__min__induct,axiom,
! [A3: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [B4: int,A6: set_int] :
( aa_set_int_o(finite_finite_int,A6)
=> ( ! [X3: int] :
( aa_set_int_o(member_int(X3),A6)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B4),X3) )
=> ( aa_set_int_o(P,A6)
=> aa_set_int_o(P,insert_int(B4,A6)) ) ) )
=> aa_set_int_o(P,A3) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7176_finite__linorder__max__induct,axiom,
! [A3: set_real,P: fun_set_real_o] :
( aa_set_real_o(finite_finite_real,A3)
=> ( aa_set_real_o(P,bot_bot_set_real)
=> ( ! [B4: real,A6: set_real] :
( aa_set_real_o(finite_finite_real,A6)
=> ( ! [X3: real] :
( aa_set_real_o(member_real(X3),A6)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B4) )
=> ( aa_set_real_o(P,A6)
=> aa_set_real_o(P,insert_real(B4,A6)) ) ) )
=> aa_set_real_o(P,A3) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7177_finite__linorder__max__induct,axiom,
! [A3: set_rat,P: fun_set_rat_o] :
( aa_set_rat_o(finite_finite_rat,A3)
=> ( aa_set_rat_o(P,bot_bot_set_rat)
=> ( ! [B4: rat,A6: set_rat] :
( aa_set_rat_o(finite_finite_rat,A6)
=> ( ! [X3: rat] :
( aa_set_rat_o(member_rat(X3),A6)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),B4) )
=> ( aa_set_rat_o(P,A6)
=> aa_set_rat_o(P,insert_rat(B4,A6)) ) ) )
=> aa_set_rat_o(P,A3) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7178_finite__linorder__max__induct,axiom,
! [A3: set_num,P: fun_set_num_o] :
( aa_set_num_o(finite_finite_num,A3)
=> ( aa_set_num_o(P,bot_bot_set_num)
=> ( ! [B4: num,A6: set_num] :
( aa_set_num_o(finite_finite_num,A6)
=> ( ! [X3: num] :
( aa_set_num_o(member_num(X3),A6)
=> aa_num_o(aa_num_fun_num_o(ord_less_num,X3),B4) )
=> ( aa_set_num_o(P,A6)
=> aa_set_num_o(P,insert_num(B4,A6)) ) ) )
=> aa_set_num_o(P,A3) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7179_finite__linorder__max__induct,axiom,
! [A3: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [B4: nat,A6: set_nat] :
( aa_set_nat_o(finite_finite_nat,A6)
=> ( ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),A6)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),B4) )
=> ( aa_set_nat_o(P,A6)
=> aa_set_nat_o(P,insert_nat(B4,A6)) ) ) )
=> aa_set_nat_o(P,A3) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7180_finite__linorder__max__induct,axiom,
! [A3: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [B4: int,A6: set_int] :
( aa_set_int_o(finite_finite_int,A6)
=> ( ! [X3: int] :
( aa_set_int_o(member_int(X3),A6)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X3),B4) )
=> ( aa_set_int_o(P,A6)
=> aa_set_int_o(P,insert_int(B4,A6)) ) ) )
=> aa_set_int_o(P,A3) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7181_sum_Oinsert__if,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A3)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),groups2240296850493347238T_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7182_sum_Oinsert__if,axiom,
! [A3: set_real,G: fun_real_real,Xb: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups8097168146408367636l_real(G,insert_real(Xb,A3)) = $ite(aa_set_real_o(member_real(Xb),A3),groups8097168146408367636l_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xb)),groups8097168146408367636l_real(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7183_sum_Oinsert__if,axiom,
! [A3: set_int,G: fun_int_real,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups8778361861064173332t_real(G,insert_int(Xb,A3)) = $ite(aa_set_int_o(member_int(Xb),A3),groups8778361861064173332t_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xb)),groups8778361861064173332t_real(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7184_sum_Oinsert__if,axiom,
! [A3: set_complex,G: fun_complex_real,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5808333547571424918x_real(G,insert_complex(Xb,A3)) = $ite(aa_set_complex_o(member_complex(Xb),A3),groups5808333547571424918x_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7185_sum_Oinsert__if,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A3)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),groups136491112297645522BT_rat(G,A3),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7186_sum_Oinsert__if,axiom,
! [A3: set_real,G: fun_real_rat,Xb: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups1300246762558778688al_rat(G,insert_real(Xb,A3)) = $ite(aa_set_real_o(member_real(Xb),A3),groups1300246762558778688al_rat(G,A3),aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xb)),groups1300246762558778688al_rat(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7187_sum_Oinsert__if,axiom,
! [A3: set_nat,G: fun_nat_rat,Xb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups2906978787729119204at_rat(G,insert_nat(Xb,A3)) = $ite(aa_set_nat_o(member_nat(Xb),A3),groups2906978787729119204at_rat(G,A3),aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xb)),groups2906978787729119204at_rat(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7188_sum_Oinsert__if,axiom,
! [A3: set_int,G: fun_int_rat,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups3906332499630173760nt_rat(G,insert_int(Xb,A3)) = $ite(aa_set_int_o(member_int(Xb),A3),groups3906332499630173760nt_rat(G,A3),aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7189_sum_Oinsert__if,axiom,
! [A3: set_complex,G: fun_complex_rat,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A3)) = $ite(aa_set_complex_o(member_complex(Xb),A3),groups5058264527183730370ex_rat(G,A3),aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7190_sum_Oinsert__if,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A3)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),groups771621172384141258BT_nat(G,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,A3))) ) ) ).
% sum.insert_if
tff(fact_7191_mult__inc,axiom,
! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Y)),Xb) ).
% mult_inc
tff(fact_7192_set__update__subset__insert,axiom,
! [Xs: list_real,I: nat,Xb: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb))),insert_real(Xb,set_real2(Xs))) ).
% set_update_subset_insert
tff(fact_7193_set__update__subset__insert,axiom,
! [Xs: list_int,I: nat,Xb: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),insert_int(Xb,set_int2(Xs))) ).
% set_update_subset_insert
tff(fact_7194_set__update__subset__insert,axiom,
! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb))),insert_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))) ).
% set_update_subset_insert
tff(fact_7195_set__update__subset__insert,axiom,
! [Xs: list_nat,I: nat,Xb: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xb))),insert_nat(Xb,set_nat2(Xs))) ).
% set_update_subset_insert
tff(fact_7196_numeral__inc,axiom,
! [Xb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).
% numeral_inc
tff(fact_7197_numeral__inc,axiom,
! [Xb: num] : numera1916890842035813515d_enat(aa_num_num(inc,Xb)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xb),one_on7984719198319812577d_enat) ).
% numeral_inc
tff(fact_7198_numeral__inc,axiom,
! [Xb: num] : numera6690914467698888265omplex(aa_num_num(inc,Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).
% numeral_inc
tff(fact_7199_numeral__inc,axiom,
! [Xb: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).
% numeral_inc
tff(fact_7200_numeral__inc,axiom,
! [Xb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).
% numeral_inc
tff(fact_7201_numeral__inc,axiom,
! [Xb: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).
% numeral_inc
tff(fact_7202_finite__induct__select,axiom,
! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
=> ( ! [T: set_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(T),S3)
=> ( aa_set_VEBT_VEBT_o(P,T)
=> ? [X3: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),minus_5127226145743854075T_VEBT(S3,T))
& aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,T)) ) ) )
=> aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).
% finite_induct_select
tff(fact_7203_finite__induct__select,axiom,
! [S3: set_complex,P: fun_set_complex_o] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( aa_set_complex_o(P,bot_bot_set_complex)
=> ( ! [T: set_complex] :
( aa_set_complex_o(ord_less_set_complex(T),S3)
=> ( aa_set_complex_o(P,T)
=> ? [X3: complex] :
( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(S3,T))
& aa_set_complex_o(P,insert_complex(X3,T)) ) ) )
=> aa_set_complex_o(P,S3) ) ) ) ).
% finite_induct_select
tff(fact_7204_finite__induct__select,axiom,
! [S3: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,S3)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [T: set_int] :
( aa_set_int_o(ord_less_set_int(T),S3)
=> ( aa_set_int_o(P,T)
=> ? [X3: int] :
( aa_set_int_o(member_int(X3),minus_minus_set_int(S3,T))
& aa_set_int_o(P,insert_int(X3,T)) ) ) )
=> aa_set_int_o(P,S3) ) ) ) ).
% finite_induct_select
tff(fact_7205_finite__induct__select,axiom,
! [S3: set_real,P: fun_set_real_o] :
( aa_set_real_o(finite_finite_real,S3)
=> ( aa_set_real_o(P,bot_bot_set_real)
=> ( ! [T: set_real] :
( aa_set_real_o(ord_less_set_real(T),S3)
=> ( aa_set_real_o(P,T)
=> ? [X3: real] :
( aa_set_real_o(member_real(X3),minus_minus_set_real(S3,T))
& aa_set_real_o(P,insert_real(X3,T)) ) ) )
=> aa_set_real_o(P,S3) ) ) ) ).
% finite_induct_select
tff(fact_7206_finite__induct__select,axiom,
! [S3: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [T: set_nat] :
( aa_set_nat_o(ord_less_set_nat(T),S3)
=> ( aa_set_nat_o(P,T)
=> ? [X3: nat] :
( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(S3,T))
& aa_set_nat_o(P,insert_nat(X3,T)) ) ) )
=> aa_set_nat_o(P,S3) ) ) ) ).
% finite_induct_select
tff(fact_7207_psubset__insert__iff,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,B3: set_VEBT_VEBT] :
( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),insert_VEBT_VEBT(Xb,B3))
<=> $ite(
aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B3),
aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),B3),
$ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B3),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),B3)) ) ) ).
% psubset_insert_iff
tff(fact_7208_psubset__insert__iff,axiom,
! [A3: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
( ord_le7866589430770878221at_nat(A3,insert8211810215607154385at_nat(Xb,B3))
<=> $ite(
member8440522571783428010at_nat(Xb,B3),
ord_le7866589430770878221at_nat(A3,B3),
$ite(member8440522571783428010at_nat(Xb,A3),ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(A3,insert8211810215607154385at_nat(Xb,bot_bo2099793752762293965at_nat)),B3),ord_le3146513528884898305at_nat(A3,B3)) ) ) ).
% psubset_insert_iff
tff(fact_7209_psubset__insert__iff,axiom,
! [A3: set_complex,Xb: complex,B3: set_complex] :
( aa_set_complex_o(ord_less_set_complex(A3),insert_complex(Xb,B3))
<=> $ite(
aa_set_complex_o(member_complex(Xb),B3),
aa_set_complex_o(ord_less_set_complex(A3),B3),
$ite(aa_set_complex_o(member_complex(Xb),A3),aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex))),B3),aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)) ) ) ).
% psubset_insert_iff
tff(fact_7210_psubset__insert__iff,axiom,
! [A3: set_int,Xb: int,B3: set_int] :
( aa_set_int_o(ord_less_set_int(A3),insert_int(Xb,B3))
<=> $ite(
aa_set_int_o(member_int(Xb),B3),
aa_set_int_o(ord_less_set_int(A3),B3),
$ite(aa_set_int_o(member_int(Xb),A3),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int))),B3),aa_set_int_o(ord_less_eq_set_int(A3),B3)) ) ) ).
% psubset_insert_iff
tff(fact_7211_psubset__insert__iff,axiom,
! [A3: set_real,Xb: real,B3: set_real] :
( aa_set_real_o(ord_less_set_real(A3),insert_real(Xb,B3))
<=> $ite(
aa_set_real_o(member_real(Xb),B3),
aa_set_real_o(ord_less_set_real(A3),B3),
$ite(aa_set_real_o(member_real(Xb),A3),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real))),B3),aa_set_real_o(ord_less_eq_set_real(A3),B3)) ) ) ).
% psubset_insert_iff
tff(fact_7212_psubset__insert__iff,axiom,
! [A3: set_nat,Xb: nat,B3: set_nat] :
( aa_set_nat_o(ord_less_set_nat(A3),insert_nat(Xb,B3))
<=> $ite(
aa_set_nat_o(member_nat(Xb),B3),
aa_set_nat_o(ord_less_set_nat(A3),B3),
$ite(aa_set_nat_o(member_nat(Xb),A3),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat))),B3),aa_set_nat_o(ord_less_eq_set_nat(A3),B3)) ) ) ).
% psubset_insert_iff
tff(fact_7213_set__replicate__Suc,axiom,
! [Nb: nat,Xb: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ).
% set_replicate_Suc
tff(fact_7214_set__replicate__Suc,axiom,
! [Nb: nat,Xb: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),Xb)) = insert_nat(Xb,bot_bot_set_nat) ).
% set_replicate_Suc
tff(fact_7215_set__replicate__Suc,axiom,
! [Nb: nat,Xb: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),Xb)) = insert_int(Xb,bot_bot_set_int) ).
% set_replicate_Suc
tff(fact_7216_set__replicate__Suc,axiom,
! [Nb: nat,Xb: real] : set_real2(replicate_real(aa_nat_nat(suc,Nb),Xb)) = insert_real(Xb,bot_bot_set_real) ).
% set_replicate_Suc
tff(fact_7217_set__replicate__conv__if,axiom,
! [Nb: nat,Xb: vEBT_VEBT] :
set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)) ).
% set_replicate_conv_if
tff(fact_7218_set__replicate__conv__if,axiom,
! [Nb: nat,Xb: nat] :
set_nat2(replicate_nat(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat(Xb,bot_bot_set_nat)) ).
% set_replicate_conv_if
tff(fact_7219_set__replicate__conv__if,axiom,
! [Nb: nat,Xb: int] :
set_int2(replicate_int(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int(Xb,bot_bot_set_int)) ).
% set_replicate_conv_if
tff(fact_7220_set__replicate__conv__if,axiom,
! [Nb: nat,Xb: real] :
set_real2(replicate_real(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_real,insert_real(Xb,bot_bot_set_real)) ).
% set_replicate_conv_if
tff(fact_7221_simp__from__to,axiom,
! [I: int,J: int] :
set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).
% simp_from_to
tff(fact_7222_sum_Oinsert__remove,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_7223_sum_Oinsert__remove,axiom,
! [A3: set_complex,G: fun_complex_real,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5808333547571424918x_real(G,insert_complex(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_7224_sum_Oinsert__remove,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_7225_sum_Oinsert__remove,axiom,
! [A3: set_complex,G: fun_complex_rat,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5058264527183730370ex_rat(G,insert_complex(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_7226_sum_Oinsert__remove,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(Xb,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_7227_sum_Oinsert__remove,axiom,
! [A3: set_complex,G: fun_complex_nat,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5693394587270226106ex_nat(G,insert_complex(Xb,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_7228_sum_Oinsert__remove,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups769130701875090982BT_int(G,insert_VEBT_VEBT(Xb,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xb)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_7229_sum_Oinsert__remove,axiom,
! [A3: set_complex,G: fun_complex_int,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups5690904116761175830ex_int(G,insert_complex(Xb,A3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_7230_sum_Oinsert__remove,axiom,
! [A3: set_int,G: fun_int_real,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups8778361861064173332t_real(G,insert_int(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xb)),groups8778361861064173332t_real(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ).
% sum.insert_remove
tff(fact_7231_sum_Oinsert__remove,axiom,
! [A3: set_int,G: fun_int_rat,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups3906332499630173760nt_rat(G,insert_int(Xb,A3)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ).
% sum.insert_remove
tff(fact_7232_sum_Oremove,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups2240296850493347238T_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xb)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_7233_sum_Oremove,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> ( groups5808333547571424918x_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_7234_sum_Oremove,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups136491112297645522BT_rat(G,A3) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xb)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_7235_sum_Oremove,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> ( groups5058264527183730370ex_rat(G,A3) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xb)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_7236_sum_Oremove,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups771621172384141258BT_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xb)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_7237_sum_Oremove,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> ( groups5693394587270226106ex_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_7238_sum_Oremove,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_int] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups769130701875090982BT_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xb)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_7239_sum_Oremove,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> ( groups5690904116761175830ex_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_7240_sum_Oremove,axiom,
! [A3: set_int,Xb: int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(member_int(Xb),A3)
=> ( groups8778361861064173332t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xb)),groups8778361861064173332t_real(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ) ).
% sum.remove
tff(fact_7241_sum_Oremove,axiom,
! [A3: set_int,Xb: int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(member_int(Xb),A3)
=> ( groups3906332499630173760nt_rat(G,A3) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xb)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ) ).
% sum.remove
tff(fact_7242_sum_Odelta__remove,axiom,
! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_hh(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B,A)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.delta_remove
tff(fact_7243_sum_Odelta__remove,axiom,
! [S3: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_hi(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).
% sum.delta_remove
tff(fact_7244_sum_Odelta__remove,axiom,
! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_hj(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(B,A)),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.delta_remove
tff(fact_7245_sum_Odelta__remove,axiom,
! [S3: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_hk(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).
% sum.delta_remove
tff(fact_7246_sum_Odelta__remove,axiom,
! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_hl(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,A)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.delta_remove
tff(fact_7247_sum_Odelta__remove,axiom,
! [S3: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_hm(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).
% sum.delta_remove
tff(fact_7248_sum_Odelta__remove,axiom,
! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_hn(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(B,A)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.delta_remove
tff(fact_7249_sum_Odelta__remove,axiom,
! [S3: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_ho(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).
% sum.delta_remove
tff(fact_7250_sum_Odelta__remove,axiom,
! [S3: set_int,A: int,B: fun_int_real,C: fun_int_real] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_hp(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(B,A)),groups8778361861064173332t_real(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups8778361861064173332t_real(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).
% sum.delta_remove
tff(fact_7251_sum_Odelta__remove,axiom,
! [S3: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_hq(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_rat_rat(plus_plus_rat(aa_int_rat(B,A)),groups3906332499630173760nt_rat(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups3906332499630173760nt_rat(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).
% sum.delta_remove
tff(fact_7252_member__le__sum,axiom,
! [I: vEBT_VEBT,A3: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A3)
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4)) )
=> ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7253_member__le__sum,axiom,
! [I: complex,A3: set_complex,F: fun_complex_rat] :
( aa_set_complex_o(member_complex(I),A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex)))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7254_member__le__sum,axiom,
! [I: int,A3: set_int,F: fun_int_rat] :
( aa_set_int_o(member_int(I),A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(A3,insert_int(I,bot_bot_set_int)))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
=> ( aa_set_int_o(finite_finite_int,A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7255_member__le__sum,axiom,
! [I: real,A3: set_real,F: fun_real_rat] :
( aa_set_real_o(member_real(I),A3)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real)))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
=> ( aa_set_real_o(finite_finite_real,A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7256_member__le__sum,axiom,
! [I: nat,A3: set_nat,F: fun_nat_rat] :
( aa_set_nat_o(member_nat(I),A3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(A3,insert_nat(I,bot_bot_set_nat)))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7257_member__le__sum,axiom,
! [I: vEBT_VEBT,A3: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A3)
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4)) )
=> ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I)),groups771621172384141258BT_nat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7258_member__le__sum,axiom,
! [I: complex,A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(member_complex(I),A3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex)))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),groups5693394587270226106ex_nat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7259_member__le__sum,axiom,
! [I: int,A3: set_int,F: fun_int_nat] :
( aa_set_int_o(member_int(I),A3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(A3,insert_int(I,bot_bot_set_int)))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
=> ( aa_set_int_o(finite_finite_int,A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),groups4541462559716669496nt_nat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7260_member__le__sum,axiom,
! [I: real,A3: set_real,F: fun_real_nat] :
( aa_set_real_o(member_real(I),A3)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real)))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
=> ( aa_set_real_o(finite_finite_real,A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),groups1935376822645274424al_nat(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7261_member__le__sum,axiom,
! [I: vEBT_VEBT,A3: set_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A3)
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_VEBT_VEBT_int(F,X4)) )
=> ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_VEBT_VEBT_int(F,I)),groups769130701875090982BT_int(F,A3)) ) ) ) ).
% member_le_sum
tff(fact_7262_unbounded__k__infinite,axiom,
! [K: nat,S3: set_nat] :
( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M2)
=> ? [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N5)
& aa_set_nat_o(member_nat(N5),S3) ) )
=> ~ aa_set_nat_o(finite_finite_nat,S3) ) ).
% unbounded_k_infinite
tff(fact_7263_infinite__nat__iff__unbounded,axiom,
! [S3: set_nat] :
( ~ aa_set_nat_o(finite_finite_nat,S3)
<=> ! [M4: nat] :
? [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N3)
& aa_set_nat_o(member_nat(N3),S3) ) ) ).
% infinite_nat_iff_unbounded
tff(fact_7264_and__nat__unfold,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
( ( Mb = zero_zero_nat )
| ( Nb = zero_zero_nat ) ),
zero_zero_nat,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).
% and_nat_unfold
tff(fact_7265_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ~ aa_set_nat_o(finite_finite_nat,S3)
<=> ! [M4: nat] :
? [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
& aa_set_nat_o(member_nat(N3),S3) ) ) ).
% infinite_nat_iff_unbounded_le
tff(fact_7266_or__Suc__0__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).
% or_Suc_0_eq
tff(fact_7267_Suc__0__or__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).
% Suc_0_or_eq
tff(fact_7268_or__nat__rec,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
| ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% or_nat_rec
tff(fact_7269_and__nat__rec,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
& ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% and_nat_rec
tff(fact_7270_or__nat__unfold,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
Mb = zero_zero_nat,
Nb,
$ite(Nb = zero_zero_nat,Mb,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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).
% or_nat_unfold
tff(fact_7271_and__int_Opelims,axiom,
! [Xb: int,Xaa: int,Y: int] :
( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
=> ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
=> ~ ( ( Y = $ite(
( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),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_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
=> ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).
% and_int.pelims
tff(fact_7272_and__int_Opsimps,axiom,
! [K: int,L: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
=> ( aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).
% and_int.psimps
tff(fact_7273_sum__gp,axiom,
! [Xb: complex,Mb: nat,Nb: nat] :
groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
zero_zero_complex,
$ite(Xb = one_one_complex,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,Nb),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ) ).
% sum_gp
tff(fact_7274_sum__gp,axiom,
! [Xb: rat,Mb: nat,Nb: nat] :
groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
zero_zero_rat,
$ite(Xb = one_one_rat,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,Nb),one_one_nat)),Mb)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ) ).
% sum_gp
tff(fact_7275_sum__gp,axiom,
! [Xb: real,Mb: nat,Nb: nat] :
groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
zero_zero_real,
$ite(Xb = one_one_real,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,Nb),one_one_nat)),Mb)),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,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ).
% sum_gp
tff(fact_7276_or__minus__numerals_I5_J,axiom,
! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),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),aa_num_num(bitM,Nb)))) ).
% or_minus_numerals(5)
tff(fact_7277_or__minus__numerals_I1_J,axiom,
! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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),aa_num_num(bitM,Nb)))) ).
% or_minus_numerals(1)
tff(fact_7278_gauss__sum__from__Suc__0,axiom,
! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% gauss_sum_from_Suc_0
tff(fact_7279_gauss__sum__from__Suc__0,axiom,
! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = 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,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum_from_Suc_0
tff(fact_7280_of__nat__eq__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
<=> ( Mb = Nb ) ) ).
% of_nat_eq_iff
tff(fact_7281_of__nat__eq__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
<=> ( Mb = Nb ) ) ).
% of_nat_eq_iff
tff(fact_7282_of__nat__eq__iff,axiom,
! [Mb: nat,Nb: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
<=> ( Mb = Nb ) ) ).
% of_nat_eq_iff
tff(fact_7283_int__eq__iff__numeral,axiom,
! [Mb: nat,V: num] :
( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
<=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).
% int_eq_iff_numeral
tff(fact_7284_abs__of__nat,axiom,
! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).
% abs_of_nat
tff(fact_7285_abs__of__nat,axiom,
! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).
% abs_of_nat
tff(fact_7286_abs__of__nat,axiom,
! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).
% abs_of_nat
tff(fact_7287_abs__of__nat,axiom,
! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).
% abs_of_nat
tff(fact_7288_of__nat__eq__0__iff,axiom,
! [Mb: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_7289_of__nat__eq__0__iff,axiom,
! [Mb: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_7290_of__nat__eq__0__iff,axiom,
! [Mb: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_7291_of__nat__eq__0__iff,axiom,
! [Mb: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_7292_of__nat__eq__0__iff,axiom,
! [Mb: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_7293_of__nat__0__eq__iff,axiom,
! [Nb: nat] :
( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
<=> ( zero_zero_nat = Nb ) ) ).
% of_nat_0_eq_iff
tff(fact_7294_of__nat__0__eq__iff,axiom,
! [Nb: nat] :
( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
<=> ( zero_zero_nat = Nb ) ) ).
% of_nat_0_eq_iff
tff(fact_7295_of__nat__0__eq__iff,axiom,
! [Nb: nat] :
( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
<=> ( zero_zero_nat = Nb ) ) ).
% of_nat_0_eq_iff
tff(fact_7296_of__nat__0__eq__iff,axiom,
! [Nb: nat] :
( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
<=> ( zero_zero_nat = Nb ) ) ).
% of_nat_0_eq_iff
tff(fact_7297_of__nat__0__eq__iff,axiom,
! [Nb: nat] :
( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
<=> ( zero_zero_nat = Nb ) ) ).
% of_nat_0_eq_iff
tff(fact_7298_of__nat__0,axiom,
aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).
% of_nat_0
tff(fact_7299_of__nat__0,axiom,
aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).
% of_nat_0
tff(fact_7300_of__nat__0,axiom,
aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).
% of_nat_0
tff(fact_7301_of__nat__0,axiom,
aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).
% of_nat_0
tff(fact_7302_of__nat__0,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).
% of_nat_0
tff(fact_7303_of__nat__less__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_iff
tff(fact_7304_of__nat__less__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_iff
tff(fact_7305_of__nat__less__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_iff
tff(fact_7306_of__nat__less__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_iff
tff(fact_7307_of__nat__numeral,axiom,
! [Nb: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,Nb)) = numera1916890842035813515d_enat(Nb) ).
% of_nat_numeral
tff(fact_7308_of__nat__numeral,axiom,
! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).
% of_nat_numeral
tff(fact_7309_of__nat__numeral,axiom,
! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).
% of_nat_numeral
tff(fact_7310_of__nat__numeral,axiom,
! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).
% of_nat_numeral
tff(fact_7311_of__nat__numeral,axiom,
! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).
% of_nat_numeral
tff(fact_7312_of__nat__le__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% of_nat_le_iff
tff(fact_7313_of__nat__le__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% of_nat_le_iff
tff(fact_7314_of__nat__le__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% of_nat_le_iff
tff(fact_7315_of__nat__le__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% of_nat_le_iff
tff(fact_7316_of__nat__add,axiom,
! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).
% of_nat_add
tff(fact_7317_of__nat__add,axiom,
! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).
% of_nat_add
tff(fact_7318_of__nat__add,axiom,
! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).
% of_nat_add
tff(fact_7319_of__nat__add,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).
% of_nat_add
tff(fact_7320_of__nat__eq__1__iff,axiom,
! [Nb: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_7321_of__nat__eq__1__iff,axiom,
! [Nb: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_7322_of__nat__eq__1__iff,axiom,
! [Nb: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_7323_of__nat__eq__1__iff,axiom,
! [Nb: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_7324_of__nat__eq__1__iff,axiom,
! [Nb: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_7325_of__nat__1__eq__iff,axiom,
! [Nb: nat] :
( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_7326_of__nat__1__eq__iff,axiom,
! [Nb: nat] :
( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_7327_of__nat__1__eq__iff,axiom,
! [Nb: nat] :
( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_7328_of__nat__1__eq__iff,axiom,
! [Nb: nat] :
( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_7329_of__nat__1__eq__iff,axiom,
! [Nb: nat] :
( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
<=> ( Nb = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_7330_of__nat__1,axiom,
aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).
% of_nat_1
tff(fact_7331_of__nat__1,axiom,
aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).
% of_nat_1
tff(fact_7332_of__nat__1,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% of_nat_1
tff(fact_7333_of__nat__1,axiom,
aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).
% of_nat_1
tff(fact_7334_of__nat__1,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).
% of_nat_1
tff(fact_7335_of__nat__mult,axiom,
! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).
% of_nat_mult
tff(fact_7336_of__nat__mult,axiom,
! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).
% of_nat_mult
tff(fact_7337_of__nat__mult,axiom,
! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).
% of_nat_mult
tff(fact_7338_of__nat__mult,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).
% of_nat_mult
tff(fact_7339_of__nat__power,axiom,
! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).
% of_nat_power
tff(fact_7340_of__nat__power,axiom,
! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).
% of_nat_power
tff(fact_7341_of__nat__power,axiom,
! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).
% of_nat_power
tff(fact_7342_of__nat__power,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).
% of_nat_power
tff(fact_7343_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xb) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7344_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7345_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7346_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7347_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
<=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7348_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
<=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7349_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
<=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7350_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
<=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7351_negative__zless,axiom,
! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).
% negative_zless
tff(fact_7352_of__nat__of__bool,axiom,
! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat((P))) = zero_n3304061248610475627l_real((P)) ).
% of_nat_of_bool
tff(fact_7353_of__nat__of__bool,axiom,
! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat((P))) = zero_n2687167440665602831ol_nat((P)) ).
% of_nat_of_bool
tff(fact_7354_of__nat__of__bool,axiom,
! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat((P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).
% of_nat_of_bool
tff(fact_7355_of__nat__of__bool,axiom,
! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,zero_n2687167440665602831ol_nat((P))) = zero_n356916108424825756nteger((P)) ).
% of_nat_of_bool
tff(fact_7356_of__nat__le__0__iff,axiom,
! [Mb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_7357_of__nat__le__0__iff,axiom,
! [Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_7358_of__nat__le__0__iff,axiom,
! [Mb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_7359_of__nat__le__0__iff,axiom,
! [Mb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
<=> ( Mb = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_7360_of__nat__Suc,axiom,
! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).
% of_nat_Suc
tff(fact_7361_of__nat__Suc,axiom,
! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).
% of_nat_Suc
tff(fact_7362_of__nat__Suc,axiom,
! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).
% of_nat_Suc
tff(fact_7363_of__nat__Suc,axiom,
! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).
% of_nat_Suc
tff(fact_7364_of__nat__Suc,axiom,
! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).
% of_nat_Suc
tff(fact_7365_numeral__less__real__of__nat__iff,axiom,
! [W: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).
% numeral_less_real_of_nat_iff
tff(fact_7366_real__of__nat__less__numeral__iff,axiom,
! [Nb: nat,W: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).
% real_of_nat_less_numeral_iff
tff(fact_7367_numeral__le__real__of__nat__iff,axiom,
! [Nb: num,Mb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).
% numeral_le_real_of_nat_iff
tff(fact_7368_of__nat__0__less__iff,axiom,
! [Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% of_nat_0_less_iff
tff(fact_7369_of__nat__0__less__iff,axiom,
! [Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% of_nat_0_less_iff
tff(fact_7370_of__nat__0__less__iff,axiom,
! [Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% of_nat_0_less_iff
tff(fact_7371_of__nat__0__less__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).
% of_nat_0_less_iff
tff(fact_7372_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_7373_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_7374_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_real_o(aa_real_fun_real_o(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,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_7375_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_7376_of__nat__power__less__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_7377_of__nat__power__less__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_7378_of__nat__power__less__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_7379_of__nat__power__less__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_7380_numeral__power__eq__of__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: nat] :
( ( power_8040749407984259932d_enat(numera1916890842035813515d_enat(Xb),Nb) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_7381_numeral__power__eq__of__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: nat] :
( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_7382_numeral__power__eq__of__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_7383_numeral__power__eq__of__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_7384_numeral__power__eq__of__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_7385_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,Xb: num,Nb: nat] :
( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y) = power_8040749407984259932d_enat(numera1916890842035813515d_enat(Xb),Nb) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7386_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,Xb: num,Nb: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7387_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,Xb: num,Nb: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7388_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,Xb: num,Nb: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7389_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,Xb: num,Nb: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7390_of__nat__power__le__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_7391_of__nat__power__le__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_7392_of__nat__power__le__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_7393_of__nat__power__le__of__nat__cancel__iff,axiom,
! [Xb: nat,B: nat,W: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_7394_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7395_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7396_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7397_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,Xb: nat] :
( aa_real_o(aa_real_fun_real_o(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,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7398_of__nat__zero__less__power__iff,axiom,
! [Xb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
| ( Nb = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7399_of__nat__zero__less__power__iff,axiom,
! [Xb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
| ( Nb = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7400_of__nat__zero__less__power__iff,axiom,
! [Xb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
| ( Nb = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7401_of__nat__zero__less__power__iff,axiom,
! [Xb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
| ( Nb = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7402_or__minus__numerals_I4_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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,Mb),aa_num_num(bit0,Nb)))) ).
% or_minus_numerals(4)
tff(fact_7403_or__minus__numerals_I8_J,axiom,
! [Nb: num,Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = 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,Mb),aa_num_num(bit0,Nb)))) ).
% or_minus_numerals(8)
tff(fact_7404_or__minus__numerals_I3_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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,Mb),aa_num_num(bitM,Nb)))) ).
% or_minus_numerals(3)
tff(fact_7405_or__minus__numerals_I7_J,axiom,
! [Nb: num,Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = 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,Mb),aa_num_num(bitM,Nb)))) ).
% or_minus_numerals(7)
tff(fact_7406_even__of__nat,axiom,
! [Nb: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).
% even_of_nat
tff(fact_7407_even__of__nat,axiom,
! [Nb: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).
% even_of_nat
tff(fact_7408_even__of__nat,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).
% even_of_nat
tff(fact_7409_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7410_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7411_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7412_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7413_of__nat__less__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7414_of__nat__less__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7415_of__nat__less__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7416_of__nat__less__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7417_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7418_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7419_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7420_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,Nb: nat,Xb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7421_of__nat__le__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7422_of__nat__le__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7423_of__nat__le__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7424_of__nat__le__numeral__power__cancel__iff,axiom,
! [Xb: nat,I: num,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7425_set__encode__insert,axiom,
! [A3: set_nat,Nb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ~ aa_set_nat_o(member_nat(Nb),A3)
=> ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A3)) = 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))),Nb)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).
% set_encode_insert
tff(fact_7426_real__arch__simple,axiom,
! [Xb: rat] :
? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% real_arch_simple
tff(fact_7427_real__arch__simple,axiom,
! [Xb: real] :
? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% real_arch_simple
tff(fact_7428_reals__Archimedean2,axiom,
! [Xb: rat] :
? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% reals_Archimedean2
tff(fact_7429_reals__Archimedean2,axiom,
! [Xb: real] :
? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% reals_Archimedean2
tff(fact_7430_mult__of__nat__commute,axiom,
! [Xb: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).
% mult_of_nat_commute
tff(fact_7431_mult__of__nat__commute,axiom,
! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).
% mult_of_nat_commute
tff(fact_7432_mult__of__nat__commute,axiom,
! [Xb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).
% mult_of_nat_commute
tff(fact_7433_mult__of__nat__commute,axiom,
! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).
% mult_of_nat_commute
tff(fact_7434_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_7435_of__nat__less__of__int__iff,axiom,
! [Nb: nat,Xb: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).
% of_nat_less_of_int_iff
tff(fact_7436_of__nat__less__of__int__iff,axiom,
! [Nb: nat,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).
% of_nat_less_of_int_iff
tff(fact_7437_of__nat__less__of__int__iff,axiom,
! [Nb: nat,Xb: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).
% of_nat_less_of_int_iff
tff(fact_7438_of__nat__0__le__iff,axiom,
! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).
% of_nat_0_le_iff
tff(fact_7439_of__nat__0__le__iff,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).
% of_nat_0_le_iff
tff(fact_7440_of__nat__0__le__iff,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).
% of_nat_0_le_iff
tff(fact_7441_of__nat__0__le__iff,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).
% of_nat_0_le_iff
tff(fact_7442_of__nat__less__0__iff,axiom,
! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).
% of_nat_less_0_iff
tff(fact_7443_of__nat__less__0__iff,axiom,
! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).
% of_nat_less_0_iff
tff(fact_7444_of__nat__less__0__iff,axiom,
! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).
% of_nat_less_0_iff
tff(fact_7445_of__nat__less__0__iff,axiom,
! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).
% of_nat_less_0_iff
tff(fact_7446_of__nat__neq__0,axiom,
! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).
% of_nat_neq_0
tff(fact_7447_of__nat__neq__0,axiom,
! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).
% of_nat_neq_0
tff(fact_7448_of__nat__neq__0,axiom,
! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).
% of_nat_neq_0
tff(fact_7449_of__nat__neq__0,axiom,
! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).
% of_nat_neq_0
tff(fact_7450_of__nat__neq__0,axiom,
! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).
% of_nat_neq_0
tff(fact_7451_div__mult2__eq_H,axiom,
! [A: int,Mb: nat,Nb: nat] : aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).
% div_mult2_eq'
tff(fact_7452_div__mult2__eq_H,axiom,
! [A: nat,Mb: nat,Nb: 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,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = 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,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).
% div_mult2_eq'
tff(fact_7453_of__nat__less__imp__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_imp_less
tff(fact_7454_of__nat__less__imp__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_imp_less
tff(fact_7455_of__nat__less__imp__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_imp_less
tff(fact_7456_of__nat__less__imp__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% of_nat_less_imp_less
tff(fact_7457_less__imp__of__nat__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).
% less_imp_of_nat_less
tff(fact_7458_less__imp__of__nat__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).
% less_imp_of_nat_less
tff(fact_7459_less__imp__of__nat__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).
% less_imp_of_nat_less
tff(fact_7460_less__imp__of__nat__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).
% less_imp_of_nat_less
tff(fact_7461_of__nat__mono,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).
% of_nat_mono
tff(fact_7462_of__nat__mono,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).
% of_nat_mono
tff(fact_7463_of__nat__mono,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).
% of_nat_mono
tff(fact_7464_of__nat__mono,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).
% of_nat_mono
tff(fact_7465_int__ops_I1_J,axiom,
aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).
% int_ops(1)
tff(fact_7466_int__ops_I3_J,axiom,
! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).
% int_ops(3)
tff(fact_7467_int__cases,axiom,
! [Z: int] :
( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
=> ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).
% int_cases
tff(fact_7468_int__of__nat__induct,axiom,
! [P: fun_int_o,Z: int] :
( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
=> ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
=> aa_int_o(P,Z) ) ) ).
% int_of_nat_induct
tff(fact_7469_nat__int__comparison_I2_J,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).
% nat_int_comparison(2)
tff(fact_7470_nat__int__comparison_I3_J,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).
% nat_int_comparison(3)
tff(fact_7471_zle__int,axiom,
! [Mb: nat,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% zle_int
tff(fact_7472_int__ops_I2_J,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% int_ops(2)
tff(fact_7473_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_7474_int__plus,axiom,
! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).
% int_plus
tff(fact_7475_zadd__int__left,axiom,
! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),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,Mb),Nb))),Z) ).
% zadd_int_left
tff(fact_7476_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_7477_not__int__zless__negative,axiom,
! [Nb: nat,Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb))) ).
% not_int_zless_negative
tff(fact_7478_of__nat__max,axiom,
! [Xb: nat,Y: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Xb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y)) ).
% of_nat_max
tff(fact_7479_of__nat__max,axiom,
! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).
% of_nat_max
tff(fact_7480_of__nat__max,axiom,
! [Xb: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).
% of_nat_max
tff(fact_7481_of__nat__max,axiom,
! [Xb: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).
% of_nat_max
tff(fact_7482_nat__less__as__int,axiom,
! [X3: nat,Xa: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).
% nat_less_as_int
tff(fact_7483_nat__leq__as__int,axiom,
! [X3: nat,Xa: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).
% nat_leq_as_int
tff(fact_7484_or__not__num__neg_Osimps_I4_J,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).
% or_not_num_neg.simps(4)
tff(fact_7485_or__not__num__neg_Osimps_I6_J,axiom,
! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).
% or_not_num_neg.simps(6)
tff(fact_7486_or__not__num__neg_Osimps_I7_J,axiom,
! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).
% or_not_num_neg.simps(7)
tff(fact_7487_or__not__num__neg_Osimps_I3_J,axiom,
! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).
% or_not_num_neg.simps(3)
tff(fact_7488_or__not__num__neg_Osimps_I5_J,axiom,
! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).
% or_not_num_neg.simps(5)
tff(fact_7489_ex__less__of__nat__mult,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
=> ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xb)) ) ).
% ex_less_of_nat_mult
tff(fact_7490_ex__less__of__nat__mult,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).
% ex_less_of_nat_mult
tff(fact_7491_of__nat__diff,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).
% of_nat_diff
tff(fact_7492_of__nat__diff,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).
% of_nat_diff
tff(fact_7493_of__nat__diff,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).
% of_nat_diff
tff(fact_7494_of__nat__diff,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).
% of_nat_diff
tff(fact_7495_exp__of__nat2__mult,axiom,
! [Xb: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).
% exp_of_nat2_mult
tff(fact_7496_exp__of__nat2__mult,axiom,
! [Xb: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).
% exp_of_nat2_mult
tff(fact_7497_exp__of__nat__mult,axiom,
! [Nb: nat,Xb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),Xb)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).
% exp_of_nat_mult
tff(fact_7498_exp__of__nat__mult,axiom,
! [Nb: nat,Xb: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).
% exp_of_nat_mult
tff(fact_7499_reals__Archimedean3,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ! [Y4: real] :
? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).
% reals_Archimedean3
tff(fact_7500_int__cases4,axiom,
! [Mb: int] :
( ! [N: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N)
=> ~ ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).
% int_cases4
tff(fact_7501_atLeast0__atMost__Suc,axiom,
! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% atLeast0_atMost_Suc
tff(fact_7502_int__Suc,axiom,
! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).
% int_Suc
tff(fact_7503_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_7504_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
<=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).
% zless_iff_Suc_zadd
tff(fact_7505_atLeastAtMost__insertL,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) ) ) ).
% atLeastAtMost_insertL
tff(fact_7506_atLeastAtMostSuc__conv,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ) ) ).
% atLeastAtMostSuc_conv
tff(fact_7507_Icc__eq__insert__lb__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) = insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) ) ) ).
% Icc_eq_insert_lb_nat
tff(fact_7508_or__not__num__neg_Osimps_I2_J,axiom,
! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).
% or_not_num_neg.simps(2)
tff(fact_7509_mod__mult2__eq_H,axiom,
! [A: int,Mb: nat,Nb: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = 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,Mb)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).
% mod_mult2_eq'
tff(fact_7510_mod__mult2__eq_H,axiom,
! [A: nat,Mb: nat,Nb: 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,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = 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,Mb)),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,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).
% mod_mult2_eq'
tff(fact_7511_or__not__num__neg_Osimps_I8_J,axiom,
! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).
% or_not_num_neg.simps(8)
tff(fact_7512_pos__int__cases,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ~ ! [N: nat] :
( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).
% pos_int_cases
tff(fact_7513_zero__less__imp__eq__int,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
=> ? [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
& ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).
% zero_less_imp_eq_int
tff(fact_7514_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N: nat] :
( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
=> ~ ! [N: nat] :
( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).
% int_cases3
tff(fact_7515_nat__less__real__le,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).
% nat_less_real_le
tff(fact_7516_nat__le__real__less,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).
% nat_le_real_less
tff(fact_7517_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,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_7518_not__zle__0__negative,axiom,
! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).
% not_zle_0_negative
tff(fact_7519_negative__zless__0,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).
% negative_zless_0
tff(fact_7520_negD,axiom,
! [Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
=> ? [N: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).
% negD
tff(fact_7521_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,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_7522_nat__approx__posE,axiom,
! [E: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
=> ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).
% nat_approx_posE
tff(fact_7523_nat__approx__posE,axiom,
! [E: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
=> ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).
% nat_approx_posE
tff(fact_7524_of__nat__less__two__power,axiom,
! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).
% of_nat_less_two_power
tff(fact_7525_of__nat__less__two__power,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% of_nat_less_two_power
tff(fact_7526_of__nat__less__two__power,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).
% of_nat_less_two_power
tff(fact_7527_inverse__of__nat__le,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( ( Nb != zero_zero_nat )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).
% inverse_of_nat_le
tff(fact_7528_inverse__of__nat__le,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( ( Nb != zero_zero_nat )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).
% inverse_of_nat_le
tff(fact_7529_exp__divide__power__eq,axiom,
! [Nb: nat,Xb: complex] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).
% exp_divide_power_eq
tff(fact_7530_exp__divide__power__eq,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( 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,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).
% exp_divide_power_eq
tff(fact_7531_real__archimedian__rdiv__eq__0,axiom,
! [Xb: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),Xb)),C) )
=> ( Xb = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
tff(fact_7532_neg__int__cases,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
=> ~ ! [N: nat] :
( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).
% neg_int_cases
tff(fact_7533_or__not__num__neg_Oelims,axiom,
! [Xb: num,Xaa: num,Y: num] :
( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( Y != one ) ) )
=> ( ( ( Xb = one )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit0,M2) )
=> ( Y != aa_num_num(bit1,M2) ) ) )
=> ( ( ( Xb = one )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit1,M2) )
=> ( Y != aa_num_num(bit1,M2) ) ) )
=> ( ( ? [N: num] : Xb = aa_num_num(bit0,N)
=> ( ( Xaa = one )
=> ( Y != aa_num_num(bit0,one) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit0,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit0,M2) )
=> ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit0,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit1,M2) )
=> ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
=> ( ( ? [N: num] : Xb = aa_num_num(bit1,N)
=> ( ( Xaa = one )
=> ( Y != one ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit1,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit0,M2) )
=> ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
=> ~ ! [N: num] :
( ( Xb = aa_num_num(bit1,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit1,M2) )
=> ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
tff(fact_7534_zdiff__int__split,axiom,
! [P: fun_int_o,Xb: nat,Y: nat] :
( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
<=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
& ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
=> aa_int_o(P,zero_zero_int) ) ) ) ).
% zdiff_int_split
tff(fact_7535_ln__realpow,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).
% ln_realpow
tff(fact_7536_linear__plus__1__le__power,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Nb)) ) ).
% linear_plus_1_le_power
tff(fact_7537_Bernoulli__inequality,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).
% Bernoulli_inequality
tff(fact_7538_set__decode__plus__power__2,axiom,
! [Nb: nat,Z: nat] :
( ~ aa_set_nat_o(member_nat(Nb),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))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).
% set_decode_plus_power_2
tff(fact_7539_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_o] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
=> ( ! [K3: int,L3: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K3),L3))
=> ( ( ~ ( aa_set_int_o(member_int(K3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& aa_set_int_o(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
=> aa_int_o(aa_int_fun_int_o(P,K3),L3) ) )
=> aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).
% and_int.pinduct
tff(fact_7540_double__arith__series,axiom,
! [A: rat,D: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_hr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).
% double_arith_series
tff(fact_7541_double__arith__series,axiom,
! [A: extended_enat,D: extended_enat,Nb: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_hs(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb),one_on7984719198319812577d_enat),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb),D))) ).
% double_arith_series
tff(fact_7542_double__arith__series,axiom,
! [A: complex,D: complex,Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ht(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),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,Nb)),D))) ).
% double_arith_series
tff(fact_7543_double__arith__series,axiom,
! [A: int,D: int,Nb: 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_hu(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = 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,Nb)),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,Nb)),D))) ).
% double_arith_series
tff(fact_7544_double__arith__series,axiom,
! [A: nat,D: nat,Nb: 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_hv(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = 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,Nb)),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,Nb)),D))) ).
% double_arith_series
tff(fact_7545_double__arith__series,axiom,
! [A: real,D: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hw(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).
% double_arith_series
tff(fact_7546_double__gauss__sum,axiom,
! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).
% double_gauss_sum
tff(fact_7547_double__gauss__sum,axiom,
! [Nb: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb),one_on7984719198319812577d_enat)) ).
% double_gauss_sum
tff(fact_7548_double__gauss__sum,axiom,
! [Nb: 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),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).
% double_gauss_sum
tff(fact_7549_double__gauss__sum,axiom,
! [Nb: 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),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).
% double_gauss_sum
tff(fact_7550_double__gauss__sum,axiom,
! [Nb: 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),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).
% double_gauss_sum
tff(fact_7551_double__gauss__sum,axiom,
! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).
% double_gauss_sum
tff(fact_7552_Bernoulli__inequality__even,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).
% Bernoulli_inequality_even
tff(fact_7553_exp__ge__one__plus__x__over__n__power__n,axiom,
! [Nb: nat,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
tff(fact_7554_exp__ge__one__minus__x__over__n__power__n,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
tff(fact_7555_double__gauss__sum__from__Suc__0,axiom,
! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_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)),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7556_double__gauss__sum__from__Suc__0,axiom,
! [Nb: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb),one_on7984719198319812577d_enat)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7557_double__gauss__sum__from__Suc__0,axiom,
! [Nb: 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)),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7558_double__gauss__sum__from__Suc__0,axiom,
! [Nb: 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)),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7559_double__gauss__sum__from__Suc__0,axiom,
! [Nb: 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)),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7560_double__gauss__sum__from__Suc__0,axiom,
! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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)),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7561_arith__series,axiom,
! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hu(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_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,Nb)),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,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% arith_series
tff(fact_7562_arith__series,axiom,
! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hv(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),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,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% arith_series
tff(fact_7563_gauss__sum,axiom,
! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% gauss_sum
tff(fact_7564_gauss__sum,axiom,
! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum
tff(fact_7565_sum__gp__offset,axiom,
! [Xb: complex,Mb: nat,Nb: nat] :
groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).
% sum_gp_offset
tff(fact_7566_sum__gp__offset,axiom,
! [Xb: rat,Mb: nat,Nb: nat] :
groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).
% sum_gp_offset
tff(fact_7567_sum__gp__offset,axiom,
! [Xb: real,Mb: nat,Nb: nat] :
groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),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,Xb),Mb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).
% sum_gp_offset
tff(fact_7568_of__nat__code__if,axiom,
! [Nb: nat] :
aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_hx,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% of_nat_code_if
tff(fact_7569_of__nat__code__if,axiom,
! [Nb: nat] :
aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = $ite(Nb = zero_zero_nat,zero_z5237406670263579293d_enat,produc2676513652042109336d_enat(aTP_Lamm_hy,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% of_nat_code_if
tff(fact_7570_of__nat__code__if,axiom,
! [Nb: nat] :
aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_hz,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% of_nat_code_if
tff(fact_7571_of__nat__code__if,axiom,
! [Nb: nat] :
aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_ia),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% of_nat_code_if
tff(fact_7572_of__nat__code__if,axiom,
! [Nb: nat] :
aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,produc1703576794950452218t_real(aTP_Lamm_ib,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% of_nat_code_if
tff(fact_7573_of__nat__code__if,axiom,
! [Nb: nat] :
aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_ic),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% of_nat_code_if
tff(fact_7574_height__double__log__univ__size,axiom,
! [U: real,Dega: nat,Ta: vEBT_VEBT] :
( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Dega) )
=> ( aa_nat_o(vEBT_invar_vebt(Ta),Dega)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).
% height_double_log_univ_size
tff(fact_7575_monoseq__arctan__series,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_id,Xb)) ) ).
% monoseq_arctan_series
tff(fact_7576_lemma__termdiff3,axiom,
! [H: real,Z: real,K5: real,Nb: nat] :
( ( H != zero_zero_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
=> aa_real_o(aa_real_fun_real_o(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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).
% lemma_termdiff3
tff(fact_7577_lemma__termdiff3,axiom,
! [H: complex,Z: complex,K5: real,Nb: nat] :
( ( H != zero_zero_complex )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5)
=> aa_real_o(aa_real_fun_real_o(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)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).
% lemma_termdiff3
tff(fact_7578_ln__series,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
=> ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ie,Xb)) ) ) ) ).
% ln_series
tff(fact_7579_zero__less__log__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).
% zero_less_log_cancel_iff
tff(fact_7580_log__less__zero__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).
% log_less_zero_cancel_iff
tff(fact_7581_one__less__log__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb) ) ) ) ).
% one_less_log_cancel_iff
tff(fact_7582_log__less__one__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A) ) ) ) ).
% log_less_one_cancel_iff
tff(fact_7583_log__less__cancel__iff,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ).
% log_less_cancel_iff
tff(fact_7584_log__eq__one,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(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_7585_log__le__cancel__iff,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).
% log_le_cancel_iff
tff(fact_7586_log__le__one__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).
% log_le_one_cancel_iff
tff(fact_7587_one__le__log__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).
% one_le_log_cancel_iff
tff(fact_7588_log__le__zero__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).
% log_le_zero_cancel_iff
tff(fact_7589_zero__le__log__cancel__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).
% zero_le_log_cancel_iff
tff(fact_7590_powser__zero,axiom,
! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_fv(F)) = aa_nat_complex(F,zero_zero_nat) ).
% powser_zero
tff(fact_7591_powser__zero,axiom,
! [F: fun_nat_real] : suminf_real(aTP_Lamm_fx(F)) = aa_nat_real(F,zero_zero_nat) ).
% powser_zero
tff(fact_7592_log__pow__cancel,axiom,
! [A: real,B: nat] :
( aa_real_o(aa_real_fun_real_o(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_7593_int__if,axiom,
! [P: $o,A: nat,B: nat] :
aa_nat_int(semiri1314217659103216013at_int,
$ite((P),A,B)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% int_if
tff(fact_7594_nat__int__comparison_I1_J,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).
% nat_int_comparison(1)
tff(fact_7595_log__of__power__eq,axiom,
! [Mb: nat,B: real,Nb: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).
% log_of_power_eq
tff(fact_7596_less__log__of__power,axiom,
! [B: real,Nb: nat,Mb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).
% less_log_of_power
tff(fact_7597_log__base__change,axiom,
! [A: real,B: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).
% log_base_change
tff(fact_7598_le__log__of__power,axiom,
! [B: real,Nb: nat,Mb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).
% le_log_of_power
tff(fact_7599_log__base__pow,axiom,
! [A: real,Nb: nat,Xb: real] :
( aa_real_o(aa_real_fun_real_o(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),Nb)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).
% log_base_pow
tff(fact_7600_log__nat__power,axiom,
! [Xb: real,B: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).
% log_nat_power
tff(fact_7601_log2__of__power__eq,axiom,
! [Mb: nat,Nb: nat] :
( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
=> ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).
% log2_of_power_eq
tff(fact_7602_log__of__power__less,axiom,
! [Mb: nat,B: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).
% log_of_power_less
tff(fact_7603_log__mult,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).
% log_mult
tff(fact_7604_log__divide,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).
% log_divide
tff(fact_7605_log__of__power__le,axiom,
! [Mb: nat,B: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).
% log_of_power_le
tff(fact_7606_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( ( B != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = 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),Xb)) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
tff(fact_7607_less__log2__of__power,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Nb)),Mb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).
% less_log2_of_power
tff(fact_7608_le__log2__of__power,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Nb)),Mb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).
% le_log2_of_power
tff(fact_7609_exp__bound__half,axiom,
! [Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).
% exp_bound_half
tff(fact_7610_exp__bound__half,axiom,
! [Z: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).
% exp_bound_half
tff(fact_7611_log2__of__power__less,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).
% log2_of_power_less
tff(fact_7612_succ__bound__size__univ_H,axiom,
! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c2(Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).
% succ_bound_size_univ'
tff(fact_7613_log2__of__power__le,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).
% log2_of_power_le
tff(fact_7614_exp__bound__lemma,axiom,
! [Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).
% exp_bound_lemma
tff(fact_7615_exp__bound__lemma,axiom,
! [Z: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).
% exp_bound_lemma
tff(fact_7616_log__base__10__eq2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(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,Xb)) ) ) ).
% log_base_10_eq2
tff(fact_7617_pred__bound__size__univ,axiom,
! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d(Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).
% pred_bound_size_univ
tff(fact_7618_insert__bound__size__univ,axiom,
! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).
% insert_bound_size_univ
tff(fact_7619_succ__bound__size__univ,axiom,
! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c(Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).
% succ_bound_size_univ
tff(fact_7620_member__bound__size__univ,axiom,
! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_m_e_m_b_e_r(Ta),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).
% member_bound_size_univ
tff(fact_7621_log__base__10__eq1,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = 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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).
% log_base_10_eq1
tff(fact_7622_arctan__series,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_if,Xb)) ) ) ).
% arctan_series
tff(fact_7623_norm__divide__numeral,axiom,
! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).
% norm_divide_numeral
tff(fact_7624_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)),aa_num_real(numeral_numeral_real,W)) ).
% norm_divide_numeral
tff(fact_7625_norm__mult__numeral1,axiom,
! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).
% norm_mult_numeral1
tff(fact_7626_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,aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).
% norm_mult_numeral1
tff(fact_7627_norm__mult__numeral2,axiom,
! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).
% norm_mult_numeral2
tff(fact_7628_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)),aa_num_real(numeral_numeral_real,W)) ).
% norm_mult_numeral2
tff(fact_7629_norm__neg__numeral,axiom,
! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).
% norm_neg_numeral
tff(fact_7630_norm__neg__numeral,axiom,
! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).
% norm_neg_numeral
tff(fact_7631_zero__less__norm__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xb))
<=> ( Xb != zero_zero_real ) ) ).
% zero_less_norm_iff
tff(fact_7632_zero__less__norm__iff,axiom,
! [Xb: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb))
<=> ( Xb != zero_zero_complex ) ) ).
% zero_less_norm_iff
tff(fact_7633_norm__one,axiom,
aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).
% norm_one
tff(fact_7634_norm__one,axiom,
aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).
% norm_one
tff(fact_7635_norm__numeral,axiom,
! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).
% norm_numeral
tff(fact_7636_norm__numeral,axiom,
! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).
% norm_numeral
tff(fact_7637_norm__not__less__zero,axiom,
! [Xb: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real) ).
% norm_not_less_zero
tff(fact_7638_norm__mult,axiom,
! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y)) ).
% norm_mult
tff(fact_7639_norm__mult,axiom,
! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).
% norm_mult
tff(fact_7640_norm__power,axiom,
! [Xb: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb) ).
% norm_power
tff(fact_7641_norm__power,axiom,
! [Xb: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb) ).
% norm_power
tff(fact_7642_norm__uminus__minus,axiom,
! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).
% norm_uminus_minus
tff(fact_7643_norm__uminus__minus,axiom,
! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).
% norm_uminus_minus
tff(fact_7644_power__eq__imp__eq__norm,axiom,
! [W: real,Nb: nat,Z: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).
% power_eq_imp_eq_norm
tff(fact_7645_power__eq__imp__eq__norm,axiom,
! [W: complex,Nb: nat,Z: complex] :
( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).
% power_eq_imp_eq_norm
tff(fact_7646_norm__mult__less,axiom,
! [Xb: real,R2: real,Y: real,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2)) ) ) ).
% norm_mult_less
tff(fact_7647_norm__mult__less,axiom,
! [Xb: complex,R2: real,Y: complex,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2)) ) ) ).
% norm_mult_less
tff(fact_7648_norm__mult__ineq,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).
% norm_mult_ineq
tff(fact_7649_norm__mult__ineq,axiom,
! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).
% norm_mult_ineq
tff(fact_7650_norm__triangle__lt,axiom,
! [Xb: real,Y: real,E: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).
% norm_triangle_lt
tff(fact_7651_norm__triangle__lt,axiom,
! [Xb: complex,Y: complex,E: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E) ) ).
% norm_triangle_lt
tff(fact_7652_norm__add__less,axiom,
! [Xb: real,R2: real,Y: real,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).
% norm_add_less
tff(fact_7653_norm__add__less,axiom,
! [Xb: complex,R2: real,Y: complex,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).
% norm_add_less
tff(fact_7654_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
=> aa_real_o(aa_real_fun_real_o(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_7655_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C)
=> aa_real_o(aa_real_fun_real_o(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_7656_norm__triangle__le,axiom,
! [Xb: real,Y: real,E: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).
% norm_triangle_le
tff(fact_7657_norm__triangle__le,axiom,
! [Xb: complex,Y: complex,E: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E) ) ).
% norm_triangle_le
tff(fact_7658_norm__triangle__ineq,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).
% norm_triangle_ineq
tff(fact_7659_norm__triangle__ineq,axiom,
! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).
% norm_triangle_ineq
tff(fact_7660_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S2)
=> aa_real_o(aa_real_fun_real_o(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,R2),S2)) ) ) ).
% norm_triangle_mono
tff(fact_7661_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S2)
=> aa_real_o(aa_real_fun_real_o(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,R2),S2)) ) ) ).
% norm_triangle_mono
tff(fact_7662_norm__power__ineq,axiom,
! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).
% norm_power_ineq
tff(fact_7663_norm__power__ineq,axiom,
! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).
% norm_power_ineq
tff(fact_7664_norm__diff__triangle__less,axiom,
! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).
% norm_diff_triangle_less
tff(fact_7665_norm__diff__triangle__less,axiom,
! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E1)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).
% norm_diff_triangle_less
tff(fact_7666_norm__diff__ineq,axiom,
! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_7667_norm__diff__ineq,axiom,
! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(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_7668_power__eq__1__iff,axiom,
! [W: real,Nb: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
=> ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
| ( Nb = zero_zero_nat ) ) ) ).
% power_eq_1_iff
tff(fact_7669_power__eq__1__iff,axiom,
! [W: complex,Nb: nat] :
( ( aa_nat_complex(power_power_complex(W),Nb) = one_one_complex )
=> ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
| ( Nb = zero_zero_nat ) ) ) ).
% power_eq_1_iff
tff(fact_7670_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(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_7671_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(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_7672_square__norm__one,axiom,
! [Xb: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
=> ( aa_real_real(real_V7735802525324610683m_real,Xb) = one_one_real ) ) ).
% square_norm_one
tff(fact_7673_square__norm__one,axiom,
! [Xb: complex] :
( ( aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
=> ( aa_complex_real(real_V1022390504157884413omplex,Xb) = one_one_real ) ) ).
% square_norm_one
tff(fact_7674_norm__power__diff,axiom,
! [Z: real,W: real,Mb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
=> aa_real_o(aa_real_fun_real_o(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),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)))) ) ) ).
% norm_power_diff
tff(fact_7675_norm__power__diff,axiom,
! [Z: complex,W: complex,Mb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),Mb)),aa_nat_complex(power_power_complex(W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W)))) ) ) ).
% norm_power_diff
tff(fact_7676_suminf__geometric,axiom,
! [C: real] :
( aa_real_o(aa_real_fun_real_o(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_7677_suminf__geometric,axiom,
! [C: complex] :
( aa_real_o(aa_real_fun_real_o(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_7678_heigt__uplog__rel,axiom,
! [Ta: vEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).
% heigt_uplog_rel
tff(fact_7679_log__ceil__idem,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
=> ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)))) ) ) ).
% log_ceil_idem
tff(fact_7680_pi__series,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ig) ).
% pi_series
tff(fact_7681_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_o] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
=> ( ! [I3: int,J2: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J2))
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
=> aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
=> aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
=> aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).
% upto.pinduct
tff(fact_7682_ceiling__numeral,axiom,
! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).
% ceiling_numeral
tff(fact_7683_ceiling__one,axiom,
aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).
% ceiling_one
tff(fact_7684_ceiling__one,axiom,
aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).
% ceiling_one
tff(fact_7685_ceiling__add__of__int,axiom,
! [Xb: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z) ).
% ceiling_add_of_int
tff(fact_7686_ceiling__add__of__int,axiom,
! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb)),Z) ).
% ceiling_add_of_int
tff(fact_7687_ceiling__le__zero,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).
% ceiling_le_zero
tff(fact_7688_ceiling__le__zero,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).
% ceiling_le_zero
tff(fact_7689_zero__less__ceiling,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).
% zero_less_ceiling
tff(fact_7690_zero__less__ceiling,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% zero_less_ceiling
tff(fact_7691_ceiling__le__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_num_rat(numeral_numeral_rat,V)) ) ).
% ceiling_le_numeral
tff(fact_7692_ceiling__le__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_num_real(numeral_numeral_real,V)) ) ).
% ceiling_le_numeral
tff(fact_7693_ceiling__less__one,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).
% ceiling_less_one
tff(fact_7694_ceiling__less__one,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).
% ceiling_less_one
tff(fact_7695_numeral__less__ceiling,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xb) ) ).
% numeral_less_ceiling
tff(fact_7696_numeral__less__ceiling,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),Xb) ) ).
% numeral_less_ceiling
tff(fact_7697_one__le__ceiling,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).
% one_le_ceiling
tff(fact_7698_one__le__ceiling,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% one_le_ceiling
tff(fact_7699_ceiling__le__one,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ).
% ceiling_le_one
tff(fact_7700_ceiling__le__one,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).
% ceiling_le_one
tff(fact_7701_one__less__ceiling,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).
% one_less_ceiling
tff(fact_7702_one__less__ceiling,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ).
% one_less_ceiling
tff(fact_7703_ceiling__add__numeral,axiom,
! [Xb: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_add_numeral
tff(fact_7704_ceiling__add__numeral,axiom,
! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_add_numeral
tff(fact_7705_ceiling__neg__numeral,axiom,
! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% ceiling_neg_numeral
tff(fact_7706_ceiling__neg__numeral,axiom,
! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% ceiling_neg_numeral
tff(fact_7707_ceiling__add__one,axiom,
! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).
% ceiling_add_one
tff(fact_7708_ceiling__add__one,axiom,
! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).
% ceiling_add_one
tff(fact_7709_ceiling__diff__numeral,axiom,
! [Xb: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_diff_numeral
tff(fact_7710_ceiling__diff__numeral,axiom,
! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_diff_numeral
tff(fact_7711_ceiling__diff__one,axiom,
! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).
% ceiling_diff_one
tff(fact_7712_ceiling__diff__one,axiom,
! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).
% ceiling_diff_one
tff(fact_7713_ceiling__numeral__power,axiom,
! [Xb: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).
% ceiling_numeral_power
tff(fact_7714_ceiling__less__zero,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).
% ceiling_less_zero
tff(fact_7715_ceiling__less__zero,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) ) ).
% ceiling_less_zero
tff(fact_7716_zero__le__ceiling,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) ) ).
% zero_le_ceiling
tff(fact_7717_zero__le__ceiling,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb) ) ).
% zero_le_ceiling
tff(fact_7718_ceiling__less__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).
% ceiling_less_numeral
tff(fact_7719_ceiling__less__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).
% ceiling_less_numeral
tff(fact_7720_numeral__le__ceiling,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xb) ) ).
% numeral_le_ceiling
tff(fact_7721_numeral__le__ceiling,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xb) ) ).
% numeral_le_ceiling
tff(fact_7722_ceiling__le__neg__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).
% ceiling_le_neg_numeral
tff(fact_7723_ceiling__le__neg__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).
% ceiling_le_neg_numeral
tff(fact_7724_neg__numeral__less__ceiling,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xb) ) ).
% neg_numeral_less_ceiling
tff(fact_7725_neg__numeral__less__ceiling,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xb) ) ).
% neg_numeral_less_ceiling
tff(fact_7726_ceiling__less__neg__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).
% ceiling_less_neg_numeral
tff(fact_7727_ceiling__less__neg__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).
% ceiling_less_neg_numeral
tff(fact_7728_neg__numeral__le__ceiling,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xb) ) ).
% neg_numeral_le_ceiling
tff(fact_7729_neg__numeral__le__ceiling,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xb) ) ).
% neg_numeral_le_ceiling
tff(fact_7730_ceiling__mono,axiom,
! [Y: rat,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,Xb)) ) ).
% ceiling_mono
tff(fact_7731_ceiling__mono,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).
% ceiling_mono
tff(fact_7732_le__of__int__ceiling,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ).
% le_of_int_ceiling
tff(fact_7733_le__of__int__ceiling,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ).
% le_of_int_ceiling
tff(fact_7734_ceiling__less__cancel,axiom,
! [Xb: rat,Y: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).
% ceiling_less_cancel
tff(fact_7735_ceiling__less__cancel,axiom,
! [Xb: real,Y: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% ceiling_less_cancel
tff(fact_7736_pi__not__less__zero,axiom,
~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).
% pi_not_less_zero
tff(fact_7737_pi__gt__zero,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).
% pi_gt_zero
tff(fact_7738_ceiling__le__iff,axiom,
! [Xb: rat,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).
% ceiling_le_iff
tff(fact_7739_ceiling__le__iff,axiom,
! [Xb: real,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).
% ceiling_le_iff
tff(fact_7740_ceiling__le,axiom,
! [Xb: rat,A: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),A) ) ).
% ceiling_le
tff(fact_7741_ceiling__le,axiom,
! [Xb: real,A: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),A) ) ).
% ceiling_le
tff(fact_7742_less__ceiling__iff,axiom,
! [Z: int,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).
% less_ceiling_iff
tff(fact_7743_less__ceiling__iff,axiom,
! [Z: int,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).
% less_ceiling_iff
tff(fact_7744_ceiling__add__le,axiom,
! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).
% ceiling_add_le
tff(fact_7745_ceiling__add__le,axiom,
! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))) ).
% ceiling_add_le
tff(fact_7746_of__int__ceiling__le__add__one,axiom,
! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat)) ).
% of_int_ceiling_le_add_one
tff(fact_7747_of__int__ceiling__le__add__one,axiom,
! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real)) ).
% of_int_ceiling_le_add_one
tff(fact_7748_of__int__ceiling__diff__one__le,axiom,
! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2) ).
% of_int_ceiling_diff_one_le
tff(fact_7749_of__int__ceiling__diff__one__le,axiom,
! [R2: real] : aa_real_o(aa_real_fun_real_o(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,R2))),one_one_real)),R2) ).
% of_int_ceiling_diff_one_le
tff(fact_7750_pi__less__4,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).
% pi_less_4
tff(fact_7751_pi__ge__two,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).
% pi_ge_two
tff(fact_7752_pi__half__neq__two,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).
% pi_half_neq_two
tff(fact_7753_ceiling__correct,axiom,
! [Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))),one_one_rat)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ) ).
% ceiling_correct
tff(fact_7754_ceiling__correct,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(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,Xb))),one_one_real)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ) ).
% ceiling_correct
tff(fact_7755_ceiling__unique,axiom,
! [Z: int,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))
=> ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = Z ) ) ) ).
% ceiling_unique
tff(fact_7756_ceiling__unique,axiom,
! [Z: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z))
=> ( aa_real_int(archim7802044766580827645g_real,Xb) = Z ) ) ) ).
% ceiling_unique
tff(fact_7757_ceiling__eq__iff,axiom,
! [Xb: rat,A: int] :
( ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = A )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).
% ceiling_eq_iff
tff(fact_7758_ceiling__eq__iff,axiom,
! [Xb: real,A: int] :
( ( aa_real_int(archim7802044766580827645g_real,Xb) = A )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A)) ) ) ).
% ceiling_eq_iff
tff(fact_7759_ceiling__split,axiom,
! [P: fun_int_o,Ta: rat] :
( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,Ta))
<=> ! [I4: int] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)),Ta)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),aa_int_rat(ring_1_of_int_rat,I4)) )
=> aa_int_o(P,I4) ) ) ).
% ceiling_split
tff(fact_7760_ceiling__split,axiom,
! [P: fun_int_o,Ta: real] :
( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
<=> ! [I4: int] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)),Ta)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),aa_int_real(ring_1_of_int_real,I4)) )
=> aa_int_o(P,I4) ) ) ).
% ceiling_split
tff(fact_7761_mult__ceiling__le,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).
% mult_ceiling_le
tff(fact_7762_mult__ceiling__le,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_int_o(aa_int_fun_int_o(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_7763_ceiling__less__iff,axiom,
! [Xb: rat,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).
% ceiling_less_iff
tff(fact_7764_ceiling__less__iff,axiom,
! [Xb: real,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_7765_le__ceiling__iff,axiom,
! [Z: int,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb) ) ).
% le_ceiling_iff
tff(fact_7766_le__ceiling__iff,axiom,
! [Z: int,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).
% le_ceiling_iff
tff(fact_7767_pi__half__neq__zero,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).
% pi_half_neq_zero
tff(fact_7768_pi__half__less__two,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% pi_half_less_two
tff(fact_7769_pi__half__le__two,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% pi_half_le_two
tff(fact_7770_ceiling__divide__upper,axiom,
! [Q3: rat,P2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),Q3)) ) ).
% ceiling_divide_upper
tff(fact_7771_ceiling__divide__upper,axiom,
! [Q3: real,P2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
=> aa_real_o(aa_real_fun_real_o(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),Q3)))),Q3)) ) ).
% ceiling_divide_upper
tff(fact_7772_pi__half__gt__zero,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% pi_half_gt_zero
tff(fact_7773_pi__half__ge__zero,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% pi_half_ge_zero
tff(fact_7774_m2pi__less__pi,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).
% m2pi_less_pi
tff(fact_7775_arctan__ubound,axiom,
! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% arctan_ubound
tff(fact_7776_arctan__one,axiom,
aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).
% arctan_one
tff(fact_7777_ceiling__divide__lower,axiom,
! [Q3: real,P2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
=> aa_real_o(aa_real_fun_real_o(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),Q3)))),one_one_real)),Q3)),P2) ) ).
% ceiling_divide_lower
tff(fact_7778_ceiling__divide__lower,axiom,
! [Q3: rat,P2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),one_one_rat)),Q3)),P2) ) ).
% ceiling_divide_lower
tff(fact_7779_ceiling__eq,axiom,
! [Nb: int,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
=> ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).
% ceiling_eq
tff(fact_7780_ceiling__eq,axiom,
! [Nb: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
=> ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).
% ceiling_eq
tff(fact_7781_minus__pi__half__less__zero,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).
% minus_pi_half_less_zero
tff(fact_7782_arctan__bounded,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).
% arctan_bounded
tff(fact_7783_arctan__lbound,axiom,
! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).
% arctan_lbound
tff(fact_7784_machin__Euler,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_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(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).
% machin_Euler
tff(fact_7785_machin,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(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,aa_num_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),aa_num_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),aa_num_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_7786_ceiling__log2__div2,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(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,Nb),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_7787_ceiling__log__nat__eq__if,axiom,
! [B: nat,Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(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,Nb),one_one_nat)))
=> ( aa_nat_o(aa_nat_fun_nat_o(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,Nb)),one_one_int) ) ) ) ) ).
% ceiling_log_nat_eq_if
tff(fact_7788_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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,Nb)),one_one_int) )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
& aa_nat_o(aa_nat_fun_nat_o(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,Nb),one_one_nat))) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
tff(fact_7789_sin__cos__npi,axiom,
! [Nb: 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))),Nb)))),pi)),aa_num_real(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)),Nb) ).
% sin_cos_npi
tff(fact_7790_lemma__termdiff2,axiom,
! [H: complex,Z: complex,Nb: 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)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ii(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).
% lemma_termdiff2
tff(fact_7791_lemma__termdiff2,axiom,
! [H: rat,Z: rat,Nb: nat] :
( ( H != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),Nb)),aa_nat_rat(power_power_rat(Z),Nb))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_ik(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).
% lemma_termdiff2
tff(fact_7792_lemma__termdiff2,axiom,
! [H: real,Z: real,Nb: 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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_im(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).
% lemma_termdiff2
tff(fact_7793_summable__arctan__series,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_if,Xb)) ) ).
% summable_arctan_series
tff(fact_7794_cos__pi__eq__zero,axiom,
! [Mb: 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))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).
% cos_pi_eq_zero
tff(fact_7795_ceiling__log__eq__powr__iff,axiom,
! [Xb: real,B: real,K: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_7796_lessThan__iff,axiom,
! [I: rat,K: rat] :
( aa_set_rat_o(member_rat(I),set_ord_lessThan_rat(K))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K) ) ).
% lessThan_iff
tff(fact_7797_lessThan__iff,axiom,
! [I: num,K: num] :
( aa_set_num_o(member_num(I),set_ord_lessThan_num(K))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K) ) ).
% lessThan_iff
tff(fact_7798_lessThan__iff,axiom,
! [I: int,K: int] :
( aa_set_int_o(member_int(I),set_ord_lessThan_int(K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).
% lessThan_iff
tff(fact_7799_lessThan__iff,axiom,
! [I: nat,K: nat] :
( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).
% lessThan_iff
tff(fact_7800_lessThan__iff,axiom,
! [I: real,K: real] :
( aa_set_real_o(member_real(I),set_or5984915006950818249n_real(K))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K) ) ).
% lessThan_iff
tff(fact_7801_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_7802_summable__iff__shift,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(aa_nat_fun_nat_real(aTP_Lamm_gj(F),K))
<=> summable_real(F) ) ).
% summable_iff_shift
tff(fact_7803_cos__zero,axiom,
aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).
% cos_zero
tff(fact_7804_cos__zero,axiom,
aa_real_real(cos_real,zero_zero_real) = one_one_real ).
% cos_zero
tff(fact_7805_lessThan__subset__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(Xb)),set_ord_lessThan_rat(Y))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).
% lessThan_subset_iff
tff(fact_7806_lessThan__subset__iff,axiom,
! [Xb: num,Y: num] :
( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(Xb)),set_ord_lessThan_num(Y))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).
% lessThan_subset_iff
tff(fact_7807_lessThan__subset__iff,axiom,
! [Xb: int,Y: int] :
( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(Xb)),set_ord_lessThan_int(Y))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).
% lessThan_subset_iff
tff(fact_7808_lessThan__subset__iff,axiom,
! [Xb: nat,Y: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).
% lessThan_subset_iff
tff(fact_7809_lessThan__subset__iff,axiom,
! [Xb: real,Y: real] :
( aa_set_real_o(ord_less_eq_set_real(set_or5984915006950818249n_real(Xb)),set_or5984915006950818249n_real(Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).
% lessThan_subset_iff
tff(fact_7810_powr__zero__eq__one,axiom,
! [Xb: real] :
aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).
% powr_zero_eq_one
tff(fact_7811_powr__gt__zero,axiom,
! [Xb: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A))
<=> ( Xb != zero_zero_real ) ) ).
% powr_gt_zero
tff(fact_7812_powr__less__cancel__iff,axiom,
! [Xb: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% powr_less_cancel_iff
tff(fact_7813_summable__cmult__iff,axiom,
! [C: complex,F: fun_nat_complex] :
( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_in(C),F))
<=> ( ( C = zero_zero_complex )
| summable_complex(F) ) ) ).
% summable_cmult_iff
tff(fact_7814_summable__cmult__iff,axiom,
! [C: real,F: fun_nat_real] :
( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eu(C),F))
<=> ( ( C = zero_zero_real )
| summable_real(F) ) ) ).
% summable_cmult_iff
tff(fact_7815_sum_OlessThan__Suc,axiom,
! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).
% sum.lessThan_Suc
tff(fact_7816_sum_OlessThan__Suc,axiom,
! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).
% sum.lessThan_Suc
tff(fact_7817_sum_OlessThan__Suc,axiom,
! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = 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,Nb))),aa_nat_nat(G,Nb)) ).
% sum.lessThan_Suc
tff(fact_7818_sum_OlessThan__Suc,axiom,
! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).
% sum.lessThan_Suc
tff(fact_7819_powr__eq__one__iff,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = one_one_real )
<=> ( Xb = zero_zero_real ) ) ) ).
% powr_eq_one_iff
tff(fact_7820_powr__le__cancel__iff,axiom,
! [Xb: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).
% powr_le_cancel_iff
tff(fact_7821_numeral__powr__numeral__real,axiom,
! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).
% numeral_powr_numeral_real
tff(fact_7822_sin__cos__squared__add3,axiom,
! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).
% sin_cos_squared_add3
tff(fact_7823_sin__cos__squared__add3,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).
% sin_cos_squared_add3
tff(fact_7824_log__powr__cancel,axiom,
! [A: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(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_7825_powr__log__cancel,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) = Xb ) ) ) ) ).
% powr_log_cancel
tff(fact_7826_sin__npi2,axiom,
! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).
% sin_npi2
tff(fact_7827_sin__npi,axiom,
! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).
% sin_npi
tff(fact_7828_sin__npi__int,axiom,
! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).
% sin_npi_int
tff(fact_7829_summable__geometric__iff,axiom,
! [C: real] :
( summable_real(aa_real_fun_nat_real(power_power_real,C))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).
% summable_geometric_iff
tff(fact_7830_summable__geometric__iff,axiom,
! [C: complex] :
( summable_complex(power_power_complex(C))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).
% summable_geometric_iff
tff(fact_7831_powr__numeral,axiom,
! [Xb: real,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).
% powr_numeral
tff(fact_7832_cos__pi__half,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).
% cos_pi_half
tff(fact_7833_sin__two__pi,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).
% sin_two_pi
tff(fact_7834_sin__pi__half,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).
% sin_pi_half
tff(fact_7835_cos__two__pi,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).
% cos_two_pi
tff(fact_7836_cos__periodic,axiom,
! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).
% cos_periodic
tff(fact_7837_sin__periodic,axiom,
! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).
% sin_periodic
tff(fact_7838_cos__2pi__minus,axiom,
! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).
% cos_2pi_minus
tff(fact_7839_cos__npi,axiom,
! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).
% cos_npi
tff(fact_7840_cos__npi2,axiom,
! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).
% cos_npi2
tff(fact_7841_sin__cos__squared__add,axiom,
! [Xb: 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,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).
% sin_cos_squared_add
tff(fact_7842_sin__cos__squared__add,axiom,
! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).
% sin_cos_squared_add
tff(fact_7843_sin__cos__squared__add2,axiom,
! [Xb: 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,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).
% sin_cos_squared_add2
tff(fact_7844_sin__cos__squared__add2,axiom,
! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).
% sin_cos_squared_add2
tff(fact_7845_sin__2npi,axiom,
! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).
% sin_2npi
tff(fact_7846_cos__2npi,axiom,
! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).
% cos_2npi
tff(fact_7847_sin__2pi__minus,axiom,
! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).
% sin_2pi_minus
tff(fact_7848_sin__int__2pin,axiom,
! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).
% sin_int_2pin
tff(fact_7849_cos__int__2pin,axiom,
! [Nb: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).
% cos_int_2pin
tff(fact_7850_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).
% cos_3over2_pi
tff(fact_7851_square__powr__half,axiom,
! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).
% square_powr_half
tff(fact_7852_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).
% sin_3over2_pi
tff(fact_7853_cos__npi__int,axiom,
! [Nb: int] :
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).
% cos_npi_int
tff(fact_7854_powr__powr,axiom,
! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).
% powr_powr
tff(fact_7855_cos__one__sin__zero,axiom,
! [Xb: complex] :
( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
=> ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).
% cos_one_sin_zero
tff(fact_7856_cos__one__sin__zero,axiom,
! [Xb: real] :
( ( aa_real_real(cos_real,Xb) = one_one_real )
=> ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).
% cos_one_sin_zero
tff(fact_7857_polar__Ex,axiom,
! [Xb: real,Y: real] :
? [R3: real,A4: real] :
( ( Xb = 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_7858_sin__diff,axiom,
! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).
% sin_diff
tff(fact_7859_sin__add,axiom,
! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).
% sin_add
tff(fact_7860_summable__comparison__test,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ? [N6: nat] :
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
=> ( summable_real(G)
=> summable_real(F) ) ) ).
% summable_comparison_test
tff(fact_7861_summable__comparison__test,axiom,
! [F: fun_nat_complex,G: fun_nat_real] :
( ? [N6: nat] :
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
=> ( summable_real(G)
=> summable_complex(F) ) ) ).
% summable_comparison_test
tff(fact_7862_summable__comparison__test_H,axiom,
! [G: fun_nat_real,N4: nat,F: fun_nat_real] :
( summable_real(G)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
=> summable_real(F) ) ) ).
% summable_comparison_test'
tff(fact_7863_summable__comparison__test_H,axiom,
! [G: fun_nat_real,N4: nat,F: fun_nat_complex] :
( summable_real(G)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
=> summable_complex(F) ) ) ).
% summable_comparison_test'
tff(fact_7864_summable__mult,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_io(F),C)) ) ).
% summable_mult
tff(fact_7865_summable__mult2,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ey(F),C)) ) ).
% summable_mult2
tff(fact_7866_summable__add,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( summable_real(F)
=> ( summable_real(G)
=> summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fk(F),G)) ) ) ).
% summable_add
tff(fact_7867_summable__add,axiom,
! [F: fun_nat_nat,G: fun_nat_nat] :
( summable_nat(F)
=> ( summable_nat(G)
=> summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fj(F),G)) ) ) ).
% summable_add
tff(fact_7868_summable__add,axiom,
! [F: fun_nat_int,G: fun_nat_int] :
( summable_int(F)
=> ( summable_int(G)
=> summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ip(F),G)) ) ) ).
% summable_add
tff(fact_7869_cos__add,axiom,
! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).
% cos_add
tff(fact_7870_cos__diff,axiom,
! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).
% cos_diff
tff(fact_7871_summable__Suc__iff,axiom,
! [F: fun_nat_real] :
( summable_real(aTP_Lamm_gh(F))
<=> summable_real(F) ) ).
% summable_Suc_iff
tff(fact_7872_summable__ignore__initial__segment,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> summable_real(aa_nat_fun_nat_real(aTP_Lamm_gj(F),K)) ) ).
% summable_ignore_initial_segment
tff(fact_7873_suminf__le__const,axiom,
! [F: fun_nat_int,Xb: int] :
( summable_int(F)
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),Xb) ) ) ).
% suminf_le_const
tff(fact_7874_suminf__le__const,axiom,
! [F: fun_nat_nat,Xb: nat] :
( summable_nat(F)
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),Xb) ) ) ).
% suminf_le_const
tff(fact_7875_suminf__le__const,axiom,
! [F: fun_nat_real,Xb: real] :
( summable_real(F)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),Xb) ) ) ).
% suminf_le_const
tff(fact_7876_lessThan__def,axiom,
! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_iq,U)) ).
% lessThan_def
tff(fact_7877_lessThan__def,axiom,
! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_ir,U)) ).
% lessThan_def
tff(fact_7878_lessThan__def,axiom,
! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_is,U)) ).
% lessThan_def
tff(fact_7879_lessThan__def,axiom,
! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,U)) ).
% lessThan_def
tff(fact_7880_lessThan__def,axiom,
! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_it,U)) ).
% lessThan_def
tff(fact_7881_summableI__nonneg__bounded,axiom,
! [F: fun_nat_int,Xb: int] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
=> summable_int(F) ) ) ).
% summableI_nonneg_bounded
tff(fact_7882_summableI__nonneg__bounded,axiom,
! [F: fun_nat_nat,Xb: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
=> summable_nat(F) ) ) ).
% summableI_nonneg_bounded
tff(fact_7883_summableI__nonneg__bounded,axiom,
! [F: fun_nat_real,Xb: real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
=> summable_real(F) ) ) ).
% summableI_nonneg_bounded
tff(fact_7884_sin__double,axiom,
! [Xb: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,Xb))),aa_complex_complex(cos_complex,Xb)) ).
% sin_double
tff(fact_7885_sin__double,axiom,
! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).
% sin_double
tff(fact_7886_powser__insidea,axiom,
! [F: fun_nat_real,Xb: real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_iv(F),Z)) ) ) ).
% powser_insidea
tff(fact_7887_powser__insidea,axiom,
! [F: fun_nat_complex,Xb: complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
=> summable_real(aa_com20334259897932645t_real(aTP_Lamm_ix(F),Z)) ) ) ).
% powser_insidea
tff(fact_7888_sincos__principal__value,axiom,
! [Xb: real] :
? [Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
& ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,Xb) )
& ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,Xb) ) ) ).
% sincos_principal_value
tff(fact_7889_suminf__le,axiom,
! [F: fun_nat_nat,G: fun_nat_nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
=> ( summable_nat(F)
=> ( summable_nat(G)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).
% suminf_le
tff(fact_7890_suminf__le,axiom,
! [F: fun_nat_int,G: fun_nat_int] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
=> ( summable_int(F)
=> ( summable_int(G)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).
% suminf_le
tff(fact_7891_suminf__le,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
=> ( summable_real(F)
=> ( summable_real(G)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).
% suminf_le
tff(fact_7892_powr__non__neg,axiom,
! [A: real,Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real) ).
% powr_non_neg
tff(fact_7893_powr__less__mono2__neg,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).
% powr_less_mono2_neg
tff(fact_7894_powr__less__mono,axiom,
! [A: real,B: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).
% powr_less_mono
tff(fact_7895_powr__less__cancel,axiom,
! [Xb: real,A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).
% powr_less_cancel
tff(fact_7896_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_gj(F),K))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).
% suminf_split_initial_segment
tff(fact_7897_suminf__minus__initial__segment,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gj(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_7898_lessThan__strict__subset__iff,axiom,
! [Mb: rat,Nb: rat] :
( aa_set_rat_o(ord_less_set_rat(set_ord_lessThan_rat(Mb)),set_ord_lessThan_rat(Nb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Nb) ) ).
% lessThan_strict_subset_iff
tff(fact_7899_lessThan__strict__subset__iff,axiom,
! [Mb: num,Nb: num] :
( aa_set_num_o(ord_less_set_num(set_ord_lessThan_num(Mb)),set_ord_lessThan_num(Nb))
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).
% lessThan_strict_subset_iff
tff(fact_7900_lessThan__strict__subset__iff,axiom,
! [Mb: int,Nb: int] :
( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(Mb)),set_ord_lessThan_int(Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).
% lessThan_strict_subset_iff
tff(fact_7901_lessThan__strict__subset__iff,axiom,
! [Mb: nat,Nb: nat] :
( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% lessThan_strict_subset_iff
tff(fact_7902_lessThan__strict__subset__iff,axiom,
! [Mb: real,Nb: real] :
( aa_set_real_o(ord_less_set_real(set_or5984915006950818249n_real(Mb)),set_or5984915006950818249n_real(Nb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).
% lessThan_strict_subset_iff
tff(fact_7903_summable__mult__D,axiom,
! [C: complex,F: fun_nat_complex] :
( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_in(C),F))
=> ( ( C != zero_zero_complex )
=> summable_complex(F) ) ) ).
% summable_mult_D
tff(fact_7904_summable__mult__D,axiom,
! [C: real,F: fun_nat_real] :
( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eu(C),F))
=> ( ( C != zero_zero_real )
=> summable_real(F) ) ) ).
% summable_mult_D
tff(fact_7905_summable__zero__power,axiom,
summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).
% summable_zero_power
tff(fact_7906_summable__zero__power,axiom,
summable_int(power_power_int(zero_zero_int)) ).
% summable_zero_power
tff(fact_7907_summable__zero__power,axiom,
summable_complex(power_power_complex(zero_zero_complex)) ).
% summable_zero_power
tff(fact_7908_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_7909_sum__less__suminf,axiom,
! [F: fun_nat_int,Nb: nat] :
( summable_int(F)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).
% sum_less_suminf
tff(fact_7910_sum__less__suminf,axiom,
! [F: fun_nat_nat,Nb: nat] :
( summable_nat(F)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).
% sum_less_suminf
tff(fact_7911_sum__less__suminf,axiom,
! [F: fun_nat_real,Nb: nat] :
( summable_real(F)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M2)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).
% sum_less_suminf
tff(fact_7912_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_ey(F),C)) ) ) ).
% suminf_mult2
tff(fact_7913_suminf__mult,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_io(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),suminf_real(F)) ) ) ).
% suminf_mult
tff(fact_7914_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_fk(F),G)) ) ) ) ).
% suminf_add
tff(fact_7915_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_fj(F),G)) ) ) ) ).
% suminf_add
tff(fact_7916_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_ip(F),G)) ) ) ) ).
% suminf_add
tff(fact_7917_sin__cos__le1,axiom,
! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))))),one_one_real) ).
% sin_cos_le1
tff(fact_7918_cos__squared__eq,axiom,
! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_squared_eq
tff(fact_7919_cos__squared__eq,axiom,
! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_squared_eq
tff(fact_7920_sin__squared__eq,axiom,
! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% sin_squared_eq
tff(fact_7921_sin__squared__eq,axiom,
! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% sin_squared_eq
tff(fact_7922_sum__less__suminf2,axiom,
! [F: fun_nat_int,Nb: nat,I: nat] :
( summable_int(F)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M2)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).
% sum_less_suminf2
tff(fact_7923_sum__less__suminf2,axiom,
! [F: fun_nat_nat,Nb: nat,I: nat] :
( summable_nat(F)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M2)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ) ) ).
% sum_less_suminf2
tff(fact_7924_sum__less__suminf2,axiom,
! [F: fun_nat_real,Nb: nat,I: nat] :
( summable_real(F)
=> ( ! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M2)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).
% sum_less_suminf2
tff(fact_7925_suminf__nonneg,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).
% suminf_nonneg
tff(fact_7926_suminf__nonneg,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).
% suminf_nonneg
tff(fact_7927_suminf__nonneg,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).
% suminf_nonneg
tff(fact_7928_suminf__eq__zero__iff,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
=> ( ( suminf_nat(F) = zero_zero_nat )
<=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).
% suminf_eq_zero_iff
tff(fact_7929_suminf__eq__zero__iff,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
=> ( ( suminf_int(F) = zero_zero_int )
<=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).
% suminf_eq_zero_iff
tff(fact_7930_suminf__eq__zero__iff,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
=> ( ( suminf_real(F) = zero_zero_real )
<=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).
% suminf_eq_zero_iff
tff(fact_7931_suminf__pos,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,N))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ).
% suminf_pos
tff(fact_7932_suminf__pos,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ).
% suminf_pos
tff(fact_7933_suminf__pos,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,N))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ).
% suminf_pos
tff(fact_7934_powr__less__mono2,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).
% powr_less_mono2
tff(fact_7935_powr__mono2_H,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).
% powr_mono2'
tff(fact_7936_powr__inj,axiom,
! [A: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
<=> ( Xb = Y ) ) ) ) ).
% powr_inj
tff(fact_7937_gr__one__powr,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ) ) ).
% gr_one_powr
tff(fact_7938_powr__mult,axiom,
! [Xb: real,Y: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(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,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).
% powr_mult
tff(fact_7939_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_7940_sin__gt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).
% sin_gt_zero
tff(fact_7941_log__powr,axiom,
! [Xb: real,B: real,Y: real] :
( ( Xb != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).
% log_powr
tff(fact_7942_summable__zero__power_H,axiom,
! [F: fun_nat_complex] : summable_complex(aTP_Lamm_fv(F)) ).
% summable_zero_power'
tff(fact_7943_summable__zero__power_H,axiom,
! [F: fun_nat_real] : summable_real(aTP_Lamm_fx(F)) ).
% summable_zero_power'
tff(fact_7944_summable__zero__power_H,axiom,
! [F: fun_nat_int] : summable_int(aTP_Lamm_iy(F)) ).
% summable_zero_power'
tff(fact_7945_summable__0__powser,axiom,
! [F: fun_nat_complex] : summable_complex(aTP_Lamm_fv(F)) ).
% summable_0_powser
tff(fact_7946_summable__0__powser,axiom,
! [F: fun_nat_real] : summable_real(aTP_Lamm_fx(F)) ).
% summable_0_powser
tff(fact_7947_ln__powr,axiom,
! [Xb: real,Y: real] :
( ( Xb != zero_zero_real )
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xb)) ) ) ).
% ln_powr
tff(fact_7948_summable__powser__split__head,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iz(F),Z))
<=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z)) ) ).
% summable_powser_split_head
tff(fact_7949_summable__powser__split__head,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_ja(F),Z))
<=> summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z)) ) ).
% summable_powser_split_head
tff(fact_7950_powser__split__head_I3_J,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iz(F),Z)) ) ).
% powser_split_head(3)
tff(fact_7951_powser__split__head_I3_J,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ja(F),Z)) ) ).
% powser_split_head(3)
tff(fact_7952_summable__powser__ignore__initial__segment,axiom,
! [F: fun_nat_complex,Mb: nat,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jb(F),Mb),Z))
<=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z)) ) ).
% summable_powser_ignore_initial_segment
tff(fact_7953_summable__powser__ignore__initial__segment,axiom,
! [F: fun_nat_real,Mb: nat,Z: real] :
( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jc(F),Mb),Z))
<=> summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z)) ) ).
% summable_powser_ignore_initial_segment
tff(fact_7954_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_7955_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,aa_num_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)),aa_num_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,Z),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% cos_diff_cos
tff(fact_7956_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_7957_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,aa_num_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)),aa_num_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% sin_diff_sin
tff(fact_7958_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_7959_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,aa_num_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)),aa_num_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(minus_minus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% sin_plus_sin
tff(fact_7960_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_7961_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% cos_times_sin
tff(fact_7962_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_7963_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% sin_times_cos
tff(fact_7964_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_7965_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% sin_times_sin
tff(fact_7966_cos__double,axiom,
! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_double
tff(fact_7967_cos__double,axiom,
! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = 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,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_double
tff(fact_7968_powr__add,axiom,
! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),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,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ).
% powr_add
tff(fact_7969_summable__norm__comparison__test,axiom,
! [F: fun_nat_complex,G: fun_nat_real] :
( ? [N6: nat] :
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
=> ( summable_real(G)
=> summable_real(aTP_Lamm_jd(F)) ) ) ).
% summable_norm_comparison_test
tff(fact_7970_summable__rabs__comparison__test,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ? [N6: nat] :
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
=> ( summable_real(G)
=> summable_real(aTP_Lamm_eq(F)) ) ) ).
% summable_rabs_comparison_test
tff(fact_7971_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_7972_sum_Onat__diff__reindex,axiom,
! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_je(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sum.nat_diff_reindex
tff(fact_7973_sum_Onat__diff__reindex,axiom,
! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jf(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sum.nat_diff_reindex
tff(fact_7974_sum__diff__distrib,axiom,
! [Q: fun_real_nat,P: fun_real_nat,Nb: real] :
( ! [X4: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X4)),aa_real_nat(P,X4))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gd(Q),P),set_or5984915006950818249n_real(Nb)) ) ) ).
% sum_diff_distrib
tff(fact_7975_sum__diff__distrib,axiom,
! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4))
=> ( 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,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gf(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).
% sum_diff_distrib
tff(fact_7976_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_7977_cos__double__sin,axiom,
! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_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_7978_suminf__pos__iff,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
<=> ? [I4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4)) ) ) ) ).
% suminf_pos_iff
tff(fact_7979_suminf__pos__iff,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
<=> ? [I4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I4)) ) ) ) ).
% suminf_pos_iff
tff(fact_7980_suminf__pos__iff,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
<=> ? [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I4)) ) ) ) ).
% suminf_pos_iff
tff(fact_7981_suminf__pos2,axiom,
! [F: fun_nat_nat,I: nat] :
( summable_nat(F)
=> ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).
% suminf_pos2
tff(fact_7982_suminf__pos2,axiom,
! [F: fun_nat_int,I: nat] :
( summable_int(F)
=> ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).
% suminf_pos2
tff(fact_7983_suminf__pos2,axiom,
! [F: fun_nat_real,I: nat] :
( summable_real(F)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).
% suminf_pos2
tff(fact_7984_cos__two__neq__zero,axiom,
aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).
% cos_two_neq_zero
tff(fact_7985_powr__realpow,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb) ) ) ).
% powr_realpow
tff(fact_7986_powr__less__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).
% powr_less_iff
tff(fact_7987_less__powr__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).
% less_powr_iff
tff(fact_7988_log__less__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).
% log_less_iff
tff(fact_7989_less__log__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).
% less_log_iff
tff(fact_7990_powser__inside,axiom,
! [F: fun_nat_real,Xb: real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z)) ) ) ).
% powser_inside
tff(fact_7991_powser__inside,axiom,
! [F: fun_nat_complex,Xb: complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z)) ) ) ).
% powser_inside
tff(fact_7992_cos__monotone__0__pi,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).
% cos_monotone_0_pi
tff(fact_7993_cos__mono__less__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ) ) ).
% cos_mono_less_eq
tff(fact_7994_sin__eq__0__pi,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
=> ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
=> ( Xb = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
tff(fact_7995_complete__algebra__summable__geometric,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),one_one_real)
=> summable_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ).
% complete_algebra_summable_geometric
tff(fact_7996_complete__algebra__summable__geometric,axiom,
! [Xb: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),one_one_real)
=> summable_complex(power_power_complex(Xb)) ) ).
% complete_algebra_summable_geometric
tff(fact_7997_summable__geometric,axiom,
! [C: real] :
( aa_real_o(aa_real_fun_real_o(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_7998_summable__geometric,axiom,
! [C: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
=> summable_complex(power_power_complex(C)) ) ).
% summable_geometric
tff(fact_7999_sin__zero__pi__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),pi)
=> ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
<=> ( Xb = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
tff(fact_8000_suminf__split__head,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( suminf_real(aTP_Lamm_gh(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_8001_sin__zero__iff__int2,axiom,
! [Xb: real] :
( ( aa_real_real(sin_real,Xb) = zero_zero_real )
<=> ? [I4: int] : Xb = 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_8002_sum__pos__lt__pair,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> ( ! [D4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat)))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).
% sum_pos_lt_pair
tff(fact_8003_sincos__total__pi,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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 )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),pi)
& ( Xb = aa_real_real(cos_real,T2) )
& ( Y = aa_real_real(sin_real,T2) ) ) ) ) ).
% sincos_total_pi
tff(fact_8004_sin__cos__sqrt,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
=> ( aa_real_real(sin_real,Xb) = 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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).
% sin_cos_sqrt
tff(fact_8005_sin__expansion__lemma,axiom,
! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% sin_expansion_lemma
tff(fact_8006_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_gq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% sum.lessThan_Suc_shift
tff(fact_8007_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_gr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% sum.lessThan_Suc_shift
tff(fact_8008_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = 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_gg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% sum.lessThan_Suc_shift
tff(fact_8009_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_gh(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% sum.lessThan_Suc_shift
tff(fact_8010_sum__lessThan__telescope_H,axiom,
! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_jg(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).
% sum_lessThan_telescope'
tff(fact_8011_sum__lessThan__telescope_H,axiom,
! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_jh(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).
% sum_lessThan_telescope'
tff(fact_8012_sum__lessThan__telescope_H,axiom,
! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_ji(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).
% sum_lessThan_telescope'
tff(fact_8013_sum__lessThan__telescope,axiom,
! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_gs(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).
% sum_lessThan_telescope
tff(fact_8014_sum__lessThan__telescope,axiom,
! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_gt(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).
% sum_lessThan_telescope
tff(fact_8015_sum__lessThan__telescope,axiom,
! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_gu(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).
% sum_lessThan_telescope
tff(fact_8016_sumr__diff__mult__const2,axiom,
! [F: fun_nat_rat,Nb: nat,R2: rat] : aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jj(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sumr_diff_mult_const2
tff(fact_8017_sumr__diff__mult__const2,axiom,
! [F: fun_nat_int,Nb: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jk(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sumr_diff_mult_const2
tff(fact_8018_sumr__diff__mult__const2,axiom,
! [F: fun_nat_real,Nb: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),R2)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jl(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sumr_diff_mult_const2
tff(fact_8019_sum_OatLeast1__atMost__eq,axiom,
! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sum.atLeast1_atMost_eq
tff(fact_8020_sum_OatLeast1__atMost__eq,axiom,
! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups6591440286371151544t_real(aTP_Lamm_gh(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% sum.atLeast1_atMost_eq
tff(fact_8021_powr__minus__divide,axiom,
! [Xb: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),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,Xb),A)) ).
% powr_minus_divide
tff(fact_8022_sum__le__suminf,axiom,
! [F: fun_nat_int,I5: set_nat] :
( summable_int(F)
=> ( aa_set_nat_o(finite_finite_nat,I5)
=> ( ! [N: nat] :
( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,I5)),suminf_int(F)) ) ) ) ).
% sum_le_suminf
tff(fact_8023_sum__le__suminf,axiom,
! [F: fun_nat_nat,I5: set_nat] :
( summable_nat(F)
=> ( aa_set_nat_o(finite_finite_nat,I5)
=> ( ! [N: nat] :
( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F)) ) ) ) ).
% sum_le_suminf
tff(fact_8024_sum__le__suminf,axiom,
! [F: fun_nat_real,I5: set_nat] :
( summable_real(F)
=> ( aa_set_nat_o(finite_finite_nat,I5)
=> ( ! [N: nat] :
( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,I5)),suminf_real(F)) ) ) ) ).
% sum_le_suminf
tff(fact_8025_cos__expansion__lemma,axiom,
! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Mb))),pi)),aa_num_real(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,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).
% cos_expansion_lemma
tff(fact_8026_sin__gt__zero__02,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).
% sin_gt_zero_02
tff(fact_8027_powr__neg__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb) ) ) ).
% powr_neg_one
tff(fact_8028_powr__mult__base,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).
% powr_mult_base
tff(fact_8029_powr__le__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).
% powr_le_iff
tff(fact_8030_le__powr__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).
% le_powr_iff
tff(fact_8031_log__le__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).
% log_le_iff
tff(fact_8032_le__log__iff,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).
% le_log_iff
tff(fact_8033_cos__two__less__zero,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).
% cos_two_less_zero
tff(fact_8034_cos__is__zero,axiom,
? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
& ( aa_real_real(cos_real,X4) = zero_zero_real )
& ! [Y4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
& ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
=> ( Y4 = X4 ) ) ) ).
% cos_is_zero
tff(fact_8035_cos__two__le__zero,axiom,
aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).
% cos_two_le_zero
tff(fact_8036_cos__monotone__minus__pi__0,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).
% cos_monotone_minus_pi_0
tff(fact_8037_one__diff__power__eq,axiom,
! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq
tff(fact_8038_one__diff__power__eq,axiom,
! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq
tff(fact_8039_one__diff__power__eq,axiom,
! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = 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),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq
tff(fact_8040_one__diff__power__eq,axiom,
! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq
tff(fact_8041_power__diff__1__eq,axiom,
! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),one_one_complex)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_1_eq
tff(fact_8042_power__diff__1__eq,axiom,
! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_1_eq
tff(fact_8043_power__diff__1__eq,axiom,
! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),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,Xb),one_one_int)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_1_eq
tff(fact_8044_power__diff__1__eq,axiom,
! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_1_eq
tff(fact_8045_sincos__total__pi__half,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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 )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( Xb = aa_real_real(cos_real,T2) )
& ( Y = aa_real_real(sin_real,T2) ) ) ) ) ) ).
% sincos_total_pi_half
tff(fact_8046_sincos__total__2pi__le,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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 )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
& ( Xb = aa_real_real(cos_real,T2) )
& ( Y = aa_real_real(sin_real,T2) ) ) ) ).
% sincos_total_2pi_le
tff(fact_8047_geometric__sum,axiom,
! [Xb: complex,Nb: nat] :
( ( Xb != one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ) ) ).
% geometric_sum
tff(fact_8048_geometric__sum,axiom,
! [Xb: rat,Nb: nat] :
( ( Xb != one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ) ) ).
% geometric_sum
tff(fact_8049_geometric__sum,axiom,
! [Xb: real,Nb: nat] :
( ( Xb != one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = 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,Xb),Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ) ).
% geometric_sum
tff(fact_8050_sincos__total__2pi,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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 )
=> ~ ! [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
=> ( ( Xb = aa_real_real(cos_real,T2) )
=> ( Y != aa_real_real(sin_real,T2) ) ) ) ) ) ).
% sincos_total_2pi
tff(fact_8051_powser__split__head_I1_J,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z))
=> ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(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_iz(F),Z))),Z)) ) ) ).
% powser_split_head(1)
tff(fact_8052_powser__split__head_I1_J,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z))
=> ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_iu(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_ja(F),Z))),Z)) ) ) ).
% powser_split_head(1)
tff(fact_8053_powser__split__head_I2_J,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z))
=> ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iz(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).
% powser_split_head(2)
tff(fact_8054_powser__split__head_I2_J,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ja(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).
% powser_split_head(2)
tff(fact_8055_ln__powr__bound,axiom,
! [Xb: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),A)) ) ) ).
% ln_powr_bound
tff(fact_8056_ln__powr__bound2,axiom,
! [Xb: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xb)) ) ) ).
% ln_powr_bound2
tff(fact_8057_log__add__eq__powr,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( ( B != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).
% log_add_eq_powr
tff(fact_8058_add__log__eq__powr,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( ( B != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = 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)),Xb)) ) ) ) ) ).
% add_log_eq_powr
tff(fact_8059_suminf__exist__split,axiom,
! [R2: real,F: fun_nat_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
=> ( summable_real(F)
=> ? [N7: nat] :
! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gj(F),N5)))),R2) ) ) ) ).
% suminf_exist_split
tff(fact_8060_suminf__exist__split,axiom,
! [R2: real,F: fun_nat_complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
=> ( summable_complex(F)
=> ? [N7: nat] :
! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_jm(F),N5)))),R2) ) ) ) ).
% suminf_exist_split
tff(fact_8061_summable__partial__sum__bound,axiom,
! [F: fun_nat_complex,E: real] :
( summable_complex(F)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
=> ~ ! [N7: nat] :
~ ! [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M3)
=> ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M3),N5)))),E) ) ) ) ).
% summable_partial_sum_bound
tff(fact_8062_summable__partial__sum__bound,axiom,
! [F: fun_nat_real,E: real] :
( summable_real(F)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
=> ~ ! [N7: nat] :
~ ! [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M3)
=> ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M3),N5)))),E) ) ) ) ).
% summable_partial_sum_bound
tff(fact_8063_minus__log__eq__powr,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( ( B != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = 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)),Xb)) ) ) ) ) ).
% minus_log_eq_powr
tff(fact_8064_summable__power__series,axiom,
! [F: fun_nat_real,Z: real] :
( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
=> ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),Z)) ) ) ) ) ).
% summable_power_series
tff(fact_8065_sin__45,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% sin_45
tff(fact_8066_Abel__lemma,axiom,
! [R2: real,R0: real,A: fun_nat_complex,M6: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,R2),R0)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
=> summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_jn(R2),A)) ) ) ) ).
% Abel_lemma
tff(fact_8067_cos__45,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% cos_45
tff(fact_8068_sum__gp__strict,axiom,
! [Xb: complex,Nb: nat] :
groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).
% sum_gp_strict
tff(fact_8069_sum__gp__strict,axiom,
! [Xb: rat,Nb: nat] :
groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).
% sum_gp_strict
tff(fact_8070_sum__gp__strict,axiom,
! [Xb: real,Nb: nat] :
groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),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,Xb),Nb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).
% sum_gp_strict
tff(fact_8071_lemma__termdiff1,axiom,
! [Z: complex,H: complex,Mb: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jo(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jp(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).
% lemma_termdiff1
tff(fact_8072_lemma__termdiff1,axiom,
! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jq(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jr(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).
% lemma_termdiff1
tff(fact_8073_lemma__termdiff1,axiom,
! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_js(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_jt(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).
% lemma_termdiff1
tff(fact_8074_lemma__termdiff1,axiom,
! [Z: real,H: real,Mb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ju(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jv(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).
% lemma_termdiff1
tff(fact_8075_diff__power__eq__sum,axiom,
! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jw(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).
% diff_power_eq_sum
tff(fact_8076_diff__power__eq__sum,axiom,
! [Xb: rat,Nb: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_jx(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).
% diff_power_eq_sum
tff(fact_8077_diff__power__eq__sum,axiom,
! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jy(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).
% diff_power_eq_sum
tff(fact_8078_diff__power__eq__sum,axiom,
! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jz(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).
% diff_power_eq_sum
tff(fact_8079_power__diff__sumr2,axiom,
! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ka(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_sumr2
tff(fact_8080_power__diff__sumr2,axiom,
! [Xb: rat,Nb: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_kb(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_sumr2
tff(fact_8081_power__diff__sumr2,axiom,
! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_kc(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_sumr2
tff(fact_8082_power__diff__sumr2,axiom,
! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_kd(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% power_diff_sumr2
tff(fact_8083_powr__def,axiom,
! [Xb: real,A: real] :
aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A) = $ite(Xb = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,Xb)))) ).
% powr_def
tff(fact_8084_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_8085_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,aa_num_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)),aa_num_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(minus_minus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% cos_plus_cos
tff(fact_8086_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_8087_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% cos_times_cos
tff(fact_8088_summable__ratio__test,axiom,
! [C: real,N4: nat,F: fun_nat_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
=> summable_real(F) ) ) ).
% summable_ratio_test
tff(fact_8089_summable__ratio__test,axiom,
! [C: real,N4: nat,F: fun_nat_complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
=> ( ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
=> summable_complex(F) ) ) ).
% summable_ratio_test
tff(fact_8090_real__sum__nat__ivl__bounded2,axiom,
! [Nb: nat,F: fun_nat_rat,K5: rat,K: nat] :
( ! [P4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K5) )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
=> aa_rat_o(aa_rat_fun_rat_o(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,Nb),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_8091_real__sum__nat__ivl__bounded2,axiom,
! [Nb: nat,F: fun_nat_int,K5: int,K: nat] :
( ! [P4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K5) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
=> aa_int_o(aa_int_fun_int_o(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,Nb),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_8092_real__sum__nat__ivl__bounded2,axiom,
! [Nb: nat,F: fun_nat_nat,K5: nat,K: nat] :
( ! [P4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K5) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
=> aa_nat_o(aa_nat_fun_nat_o(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,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_8093_real__sum__nat__ivl__bounded2,axiom,
! [Nb: nat,F: fun_nat_real,K5: real,K: nat] :
( ! [P4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K5) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
=> aa_real_o(aa_real_fun_real_o(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,Nb),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_8094_sin__gt__zero2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).
% sin_gt_zero2
tff(fact_8095_sin__lt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).
% sin_lt_zero
tff(fact_8096_cos__double__less__one,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).
% cos_double_less_one
tff(fact_8097_sin__30,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(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_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% sin_30
tff(fact_8098_cos__gt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).
% cos_gt_zero
tff(fact_8099_sin__monotone__2pi__le,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).
% sin_monotone_2pi_le
tff(fact_8100_sin__mono__le__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).
% sin_mono_le_eq
tff(fact_8101_sin__inj__pi,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
=> ( Xb = Y ) ) ) ) ) ) ).
% sin_inj_pi
tff(fact_8102_log__minus__eq__powr,axiom,
! [B: real,Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( ( B != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).
% log_minus_eq_powr
tff(fact_8103_cos__60,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% cos_60
tff(fact_8104_sin__60,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% sin_60
tff(fact_8105_cos__30,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% cos_30
tff(fact_8106_cos__one__2pi__int,axiom,
! [Xb: real] :
( ( aa_real_real(cos_real,Xb) = one_one_real )
<=> ? [X: int] : Xb = 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,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).
% cos_one_2pi_int
tff(fact_8107_one__diff__power__eq_H,axiom,
! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ke(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq'
tff(fact_8108_one__diff__power__eq_H,axiom,
! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_kf(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq'
tff(fact_8109_one__diff__power__eq_H,axiom,
! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = 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),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_kg(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq'
tff(fact_8110_one__diff__power__eq_H,axiom,
! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_kh(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% one_diff_power_eq'
tff(fact_8111_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_8112_cos__double__cos,axiom,
! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_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_8113_cos__treble__cos,axiom,
! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = 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,Xb)),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,Xb))) ).
% cos_treble_cos
tff(fact_8114_cos__treble__cos,axiom,
! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).
% cos_treble_cos
tff(fact_8115_powr__half__sqrt,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).
% powr_half_sqrt
tff(fact_8116_powr__neg__numeral,axiom,
! [Xb: real,Nb: num] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).
% powr_neg_numeral
tff(fact_8117_sin__le__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).
% sin_le_zero
tff(fact_8118_sin__less__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).
% sin_less_zero
tff(fact_8119_sin__mono__less__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).
% sin_mono_less_eq
tff(fact_8120_sin__monotone__2pi,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).
% sin_monotone_2pi
tff(fact_8121_sin__total,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> ? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(sin_real,X4) = Y )
& ! [Y4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(sin_real,Y4) = Y ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% sin_total
tff(fact_8122_cos__gt__zero__pi,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).
% cos_gt_zero_pi
tff(fact_8123_cos__ge__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).
% cos_ge_zero
tff(fact_8124_cos__one__2pi,axiom,
! [Xb: real] :
( ( aa_real_real(cos_real,Xb) = one_one_real )
<=> ( ? [X: nat] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
| ? [X: nat] : Xb = 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,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).
% cos_one_2pi
tff(fact_8125_sum__split__even__odd,axiom,
! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ki(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))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_kj(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_kk(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% sum_split_even_odd
tff(fact_8126_sin__pi__divide__n__gt__0,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> aa_real_o(aa_real_fun_real_o(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,Nb)))) ) ).
% sin_pi_divide_n_gt_0
tff(fact_8127_sin__arctan,axiom,
! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sin_arctan
tff(fact_8128_cos__arctan,axiom,
! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% cos_arctan
tff(fact_8129_sin__zero__iff__int,axiom,
! [Xb: real] :
( ( aa_real_real(sin_real,Xb) = zero_zero_real )
<=> ? [I4: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4)
& ( Xb = 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).
% sin_zero_iff_int
tff(fact_8130_cos__zero__iff__int,axiom,
! [Xb: real] :
( ( aa_real_real(cos_real,Xb) = zero_zero_real )
<=> ? [I4: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4)
& ( Xb = 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).
% cos_zero_iff_int
tff(fact_8131_sin__zero__lemma,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
=> ? [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
& ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).
% sin_zero_lemma
tff(fact_8132_sin__zero__iff,axiom,
! [Xb: real] :
( ( aa_real_real(sin_real,Xb) = zero_zero_real )
<=> ( ? [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
& ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
| ? [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
& ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).
% sin_zero_iff
tff(fact_8133_cos__zero__lemma,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
=> ? [N: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
& ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).
% cos_zero_lemma
tff(fact_8134_cos__zero__iff,axiom,
! [Xb: real] :
( ( aa_real_real(cos_real,Xb) = zero_zero_real )
<=> ( ? [N3: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
& ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
| ? [N3: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
& ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).
% cos_zero_iff
tff(fact_8135_sum__bounds__lt__plus1,axiom,
! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gg(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_8136_sum__bounds__lt__plus1,axiom,
! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_gh(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_8137_sumr__cos__zero__one,axiom,
! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_kl,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).
% sumr_cos_zero_one
tff(fact_8138_tan__double,axiom,
! [Xb: complex] :
( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
=> ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% tan_double
tff(fact_8139_tan__double,axiom,
! [Xb: real] :
( ( aa_real_real(cos_real,Xb) != zero_zero_real )
=> ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
=> ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% tan_double
tff(fact_8140_sin__tan,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% sin_tan
tff(fact_8141_cos__tan,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_real(cos_real,Xb) = 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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% cos_tan
tff(fact_8142_tan__npi,axiom,
! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).
% tan_npi
tff(fact_8143_tan__periodic__n,axiom,
! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).
% tan_periodic_n
tff(fact_8144_tan__periodic__nat,axiom,
! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).
% tan_periodic_nat
tff(fact_8145_tan__periodic__int,axiom,
! [Xb: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xb) ).
% tan_periodic_int
tff(fact_8146_tan__periodic,axiom,
! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).
% tan_periodic
tff(fact_8147_tan__45,axiom,
aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).
% tan_45
tff(fact_8148_tan__60,axiom,
aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).
% tan_60
tff(fact_8149_lemma__tan__total,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
=> ? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X4)) ) ) ).
% lemma_tan_total
tff(fact_8150_tan__gt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).
% tan_gt_zero
tff(fact_8151_lemma__tan__total1,axiom,
! [Y: real] :
? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(tan_real,X4) = Y ) ) ).
% lemma_tan_total1
tff(fact_8152_tan__mono__lt__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).
% tan_mono_lt_eq
tff(fact_8153_tan__monotone_H,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).
% tan_monotone'
tff(fact_8154_tan__monotone,axiom,
! [Y: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ).
% tan_monotone
tff(fact_8155_tan__total,axiom,
! [Y: real] :
? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(tan_real,X4) = Y )
& ! [Y4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(tan_real,Y4) = Y ) )
=> ( Y4 = X4 ) ) ) ).
% tan_total
tff(fact_8156_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),aa_num_real(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_8157_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)) ).
% tan_inverse
tff(fact_8158_add__tan__eq,axiom,
! [Xb: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).
% add_tan_eq
tff(fact_8159_add__tan__eq,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(cos_real,Xb) != 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,Xb)),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,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).
% add_tan_eq
tff(fact_8160_tan__pos__pi2__le,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).
% tan_pos_pi2_le
tff(fact_8161_tan__total__pos,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ? [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(tan_real,X4) = Y ) ) ) ).
% tan_total_pos
tff(fact_8162_tan__less__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).
% tan_less_zero
tff(fact_8163_tan__mono__le__eq,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).
% tan_mono_le_eq
tff(fact_8164_tan__mono__le,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) ) ) ) ).
% tan_mono_le
tff(fact_8165_tan__bound__pi2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xb))),one_one_real) ) ).
% tan_bound_pi2
tff(fact_8166_tan__30,axiom,
aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).
% tan_30
tff(fact_8167_arctan__unique,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( ( aa_real_real(tan_real,Xb) = Y )
=> ( aa_real_real(arctan,Y) = Xb ) ) ) ) ).
% arctan_unique
tff(fact_8168_arctan__tan,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).
% arctan_tan
tff(fact_8169_arctan,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).
% arctan
tff(fact_8170_tan__add,axiom,
! [Xb: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) != zero_zero_complex )
=> ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),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,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).
% tan_add
tff(fact_8171_tan__add,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(cos_real,Xb) != 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,Xb),Y)) != zero_zero_real )
=> ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),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,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).
% tan_add
tff(fact_8172_tan__diff,axiom,
! [Xb: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) != zero_zero_complex )
=> ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xb)),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,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).
% tan_diff
tff(fact_8173_tan__diff,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(cos_real,Xb) != 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,Xb),Y)) != zero_zero_real )
=> ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,Xb)),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,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).
% tan_diff
tff(fact_8174_lemma__tan__add1,axiom,
! [Xb: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,Xb) != 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,Xb)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).
% lemma_tan_add1
tff(fact_8175_lemma__tan__add1,axiom,
! [Xb: real,Y: real] :
( ( aa_real_real(cos_real,Xb) != 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,Xb)),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,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).
% lemma_tan_add1
tff(fact_8176_tan__total__pi4,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ? [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,Z2) = Xb ) ) ) ).
% tan_total_pi4
tff(fact_8177_tan__half,axiom,
! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),one_one_complex)) ).
% tan_half
tff(fact_8178_tan__half,axiom,
! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).
% tan_half
tff(fact_8179_Maclaurin__minus__cos__expansion,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),zero_zero_real)
& ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_km,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
tff(fact_8180_Maclaurin__cos__expansion2,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Xb)
& ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_km,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).
% Maclaurin_cos_expansion2
tff(fact_8181_Maclaurin__cos__expansion,axiom,
! [Xb: real,Nb: nat] :
? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_km,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).
% Maclaurin_cos_expansion
tff(fact_8182_arcosh__def,axiom,
! [Xb: real] : aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).
% arcosh_def
tff(fact_8183_Maclaurin__exp__lt,axiom,
! [Xb: real,Nb: nat] :
( ( Xb != zero_zero_real )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kn,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).
% Maclaurin_exp_lt
tff(fact_8184_fact__0,axiom,
aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).
% fact_0
tff(fact_8185_fact__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).
% fact_0
tff(fact_8186_fact__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).
% fact_0
tff(fact_8187_fact__0,axiom,
aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).
% fact_0
tff(fact_8188_fact__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).
% fact_0
tff(fact_8189_fact__1,axiom,
aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).
% fact_1
tff(fact_8190_fact__1,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).
% fact_1
tff(fact_8191_fact__1,axiom,
aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).
% fact_1
tff(fact_8192_fact__1,axiom,
aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).
% fact_1
tff(fact_8193_fact__1,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).
% fact_1
tff(fact_8194_of__real__eq__1__iff,axiom,
! [Xb: real] :
( ( aa_real_real(real_V1803761363581548252l_real,Xb) = one_one_real )
<=> ( Xb = one_one_real ) ) ).
% of_real_eq_1_iff
tff(fact_8195_of__real__eq__1__iff,axiom,
! [Xb: real] :
( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = one_one_complex )
<=> ( Xb = one_one_real ) ) ).
% of_real_eq_1_iff
tff(fact_8196_of__real__1,axiom,
aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).
% of_real_1
tff(fact_8197_of__real__1,axiom,
aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).
% of_real_1
tff(fact_8198_of__real__numeral,axiom,
! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).
% of_real_numeral
tff(fact_8199_of__real__numeral,axiom,
! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).
% of_real_numeral
tff(fact_8200_of__real__mult,axiom,
! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).
% of_real_mult
tff(fact_8201_of__real__mult,axiom,
! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).
% of_real_mult
tff(fact_8202_of__real__add,axiom,
! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).
% of_real_add
tff(fact_8203_of__real__add,axiom,
! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).
% of_real_add
tff(fact_8204_of__real__power,axiom,
! [Xb: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,Xb)),Nb) ).
% of_real_power
tff(fact_8205_of__real__power,axiom,
! [Xb: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),Nb) ).
% of_real_power
tff(fact_8206_fact__Suc__0,axiom,
aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).
% fact_Suc_0
tff(fact_8207_fact__Suc__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).
% fact_Suc_0
tff(fact_8208_fact__Suc__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).
% fact_Suc_0
tff(fact_8209_fact__Suc__0,axiom,
aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).
% fact_Suc_0
tff(fact_8210_fact__Suc__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% fact_Suc_0
tff(fact_8211_fact__Suc,axiom,
! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).
% fact_Suc
tff(fact_8212_fact__Suc,axiom,
! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).
% fact_Suc
tff(fact_8213_fact__Suc,axiom,
! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).
% fact_Suc
tff(fact_8214_fact__Suc,axiom,
! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).
% fact_Suc
tff(fact_8215_fact__2,axiom,
semiri4449623510593786356d_enat(aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).
% fact_2
tff(fact_8216_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_8217_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_8218_fact__2,axiom,
aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).
% fact_2
tff(fact_8219_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_8220_of__real__neg__numeral,axiom,
! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) ).
% of_real_neg_numeral
tff(fact_8221_of__real__neg__numeral,axiom,
! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).
% of_real_neg_numeral
tff(fact_8222_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_8223_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_8224_norm__of__real__add1,axiom,
! [Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).
% norm_of_real_add1
tff(fact_8225_norm__of__real__add1,axiom,
! [Xb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).
% norm_of_real_add1
tff(fact_8226_norm__of__real__addn,axiom,
! [Xb: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_num_real(numeral_numeral_real,B))) ).
% norm_of_real_addn
tff(fact_8227_norm__of__real__addn,axiom,
! [Xb: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_num_real(numeral_numeral_real,B))) ).
% norm_of_real_addn
tff(fact_8228_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).
% cos_of_real_pi_half
tff(fact_8229_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_8230_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).
% sin_of_real_pi_half
tff(fact_8231_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_8232_fact__ge__zero,axiom,
! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).
% fact_ge_zero
tff(fact_8233_fact__ge__zero,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).
% fact_ge_zero
tff(fact_8234_fact__ge__zero,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).
% fact_ge_zero
tff(fact_8235_fact__ge__zero,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).
% fact_ge_zero
tff(fact_8236_fact__gt__zero,axiom,
! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).
% fact_gt_zero
tff(fact_8237_fact__gt__zero,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).
% fact_gt_zero
tff(fact_8238_fact__gt__zero,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).
% fact_gt_zero
tff(fact_8239_fact__gt__zero,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).
% fact_gt_zero
tff(fact_8240_fact__not__neg,axiom,
! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).
% fact_not_neg
tff(fact_8241_fact__not__neg,axiom,
! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).
% fact_not_neg
tff(fact_8242_fact__not__neg,axiom,
! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).
% fact_not_neg
tff(fact_8243_fact__not__neg,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).
% fact_not_neg
tff(fact_8244_fact__ge__1,axiom,
! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).
% fact_ge_1
tff(fact_8245_fact__ge__1,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).
% fact_ge_1
tff(fact_8246_fact__ge__1,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).
% fact_ge_1
tff(fact_8247_fact__ge__1,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).
% fact_ge_1
tff(fact_8248_fact__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).
% fact_mono
tff(fact_8249_fact__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).
% fact_mono
tff(fact_8250_fact__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).
% fact_mono
tff(fact_8251_fact__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).
% fact_mono
tff(fact_8252_fact__dvd,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).
% fact_dvd
tff(fact_8253_fact__dvd,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).
% fact_dvd
tff(fact_8254_fact__dvd,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).
% fact_dvd
tff(fact_8255_fact__dvd,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).
% fact_dvd
tff(fact_8256_fact__less__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).
% fact_less_mono
tff(fact_8257_fact__less__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).
% fact_less_mono
tff(fact_8258_fact__less__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).
% fact_less_mono
tff(fact_8259_fact__less__mono,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).
% fact_less_mono
tff(fact_8260_fact__fact__dvd__fact,axiom,
! [K: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).
% fact_fact_dvd_fact
tff(fact_8261_fact__fact__dvd__fact,axiom,
! [K: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).
% fact_fact_dvd_fact
tff(fact_8262_fact__fact__dvd__fact,axiom,
! [K: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(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,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).
% fact_fact_dvd_fact
tff(fact_8263_fact__fact__dvd__fact,axiom,
! [K: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).
% fact_fact_dvd_fact
tff(fact_8264_fact__fact__dvd__fact,axiom,
! [K: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(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,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).
% fact_fact_dvd_fact
tff(fact_8265_fact__mod,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).
% fact_mod
tff(fact_8266_fact__mod,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).
% fact_mod
tff(fact_8267_fact__le__power,axiom,
! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).
% fact_le_power
tff(fact_8268_fact__le__power,axiom,
! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).
% fact_le_power
tff(fact_8269_fact__le__power,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).
% fact_le_power
tff(fact_8270_fact__le__power,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).
% fact_le_power
tff(fact_8271_norm__less__p1,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),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,Xb))),one_one_real))) ).
% norm_less_p1
tff(fact_8272_norm__less__p1,axiom,
! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xb))),one_one_complex))) ).
% norm_less_p1
tff(fact_8273_choose__dvd,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(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,Nb),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).
% choose_dvd
tff(fact_8274_choose__dvd,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> aa_rat_o(aa_rat_fun_rat_o(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,Nb),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).
% choose_dvd
tff(fact_8275_choose__dvd,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> aa_int_o(aa_int_fun_int_o(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,Nb),K)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).
% choose_dvd
tff(fact_8276_choose__dvd,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> aa_real_o(aa_real_fun_real_o(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,Nb),K)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).
% choose_dvd
tff(fact_8277_choose__dvd,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(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,Nb),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).
% choose_dvd
tff(fact_8278_fact__numeral,axiom,
! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8279_fact__numeral,axiom,
! [K: num] : semiri4449623510593786356d_enat(aa_num_nat(numeral_numeral_nat,K)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(K),semiri4449623510593786356d_enat(aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8280_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_8281_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_8282_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,aa_num_real(numeral_numeral_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8283_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_8284_square__fact__le__2__fact,axiom,
! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),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))),Nb))) ).
% square_fact_le_2_fact
tff(fact_8285_cos__int__times__real,axiom,
! [Mb: int,Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),aa_real_real(real_V1803761363581548252l_real,Xb))) = 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,Mb)),Xb))) ).
% cos_int_times_real
tff(fact_8286_cos__int__times__real,axiom,
! [Mb: int,Xb: real] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Mb)),aa_real_complex(real_V4546457046886955230omplex,Xb))) = 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,Mb)),Xb))) ).
% cos_int_times_real
tff(fact_8287_sin__int__times__real,axiom,
! [Mb: int,Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),aa_real_real(real_V1803761363581548252l_real,Xb))) = 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,Mb)),Xb))) ).
% sin_int_times_real
tff(fact_8288_sin__int__times__real,axiom,
! [Mb: int,Xb: real] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Mb)),aa_real_complex(real_V4546457046886955230omplex,Xb))) = 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,Mb)),Xb))) ).
% sin_int_times_real
tff(fact_8289_fact__num__eq__if,axiom,
! [Mb: nat] :
aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_8290_fact__num__eq__if,axiom,
! [Mb: nat] :
aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_8291_fact__num__eq__if,axiom,
! [Mb: nat] :
aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_8292_fact__num__eq__if,axiom,
! [Mb: nat] :
aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_8293_fact__num__eq__if,axiom,
! [Mb: nat] :
aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_8294_fact__reduce,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8295_fact__reduce,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8296_fact__reduce,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8297_fact__reduce,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8298_cos__sin__eq,axiom,
! [Xb: real] : aa_real_real(cos_real,Xb) = 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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) ).
% cos_sin_eq
tff(fact_8299_cos__sin__eq,axiom,
! [Xb: complex] : aa_complex_complex(cos_complex,Xb) = 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)))),Xb)) ).
% cos_sin_eq
tff(fact_8300_sin__cos__eq,axiom,
! [Xb: real] : aa_real_real(sin_real,Xb) = 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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) ).
% sin_cos_eq
tff(fact_8301_sin__cos__eq,axiom,
! [Xb: complex] : aa_complex_complex(sin_complex,Xb) = 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)))),Xb)) ).
% sin_cos_eq
tff(fact_8302_Maclaurin__zero,axiom,
! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
( ( Xb = zero_zero_real )
=> ( ( Nb != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_ko(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).
% Maclaurin_zero
tff(fact_8303_Maclaurin__zero,axiom,
! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
( ( Xb = zero_zero_real )
=> ( ( Nb != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kp(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).
% Maclaurin_zero
tff(fact_8304_Maclaurin__zero,axiom,
! [Xb: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
( ( Xb = zero_zero_real )
=> ( ( Nb != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_kq(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).
% Maclaurin_zero
tff(fact_8305_Maclaurin__zero,axiom,
! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
( ( Xb = zero_zero_real )
=> ( ( Nb != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_kr(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).
% Maclaurin_zero
tff(fact_8306_Maclaurin__zero,axiom,
! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
( ( Xb = zero_zero_real )
=> ( ( Nb != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_ks(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).
% Maclaurin_zero
tff(fact_8307_Maclaurin__lemma,axiom,
! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(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_kt(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).
% Maclaurin_lemma
tff(fact_8308_minus__sin__cos__eq,axiom,
! [Xb: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% minus_sin_cos_eq
tff(fact_8309_minus__sin__cos__eq,axiom,
! [Xb: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,Xb)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).
% minus_sin_cos_eq
tff(fact_8310_Maclaurin__exp__le,axiom,
! [Xb: real,Nb: nat] :
? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kn,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).
% Maclaurin_exp_le
tff(fact_8311_cos__coeff__def,axiom,
! [X3: nat] :
aa_nat_real(cos_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),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,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3)),zero_zero_real) ).
% cos_coeff_def
tff(fact_8312_arsinh__def,axiom,
! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).
% arsinh_def
tff(fact_8313_Maclaurin__sin__expansion3,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Xb)
& ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ku,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).
% Maclaurin_sin_expansion3
tff(fact_8314_Maclaurin__sin__expansion4,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),Xb)
& ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ku,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).
% Maclaurin_sin_expansion4
tff(fact_8315_Maclaurin__sin__expansion2,axiom,
! [Xb: real,Nb: nat] :
? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ku,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).
% Maclaurin_sin_expansion2
tff(fact_8316_Maclaurin__sin__expansion,axiom,
! [Xb: real,Nb: nat] :
? [T2: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ku,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ).
% Maclaurin_sin_expansion
tff(fact_8317_sin__coeff__def,axiom,
! [X3: nat] :
aa_nat_real(sin_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),zero_zero_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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3))) ).
% sin_coeff_def
tff(fact_8318_fact__ge__self,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).
% fact_ge_self
tff(fact_8319_fact__mono__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).
% fact_mono_nat
tff(fact_8320_fact__less__mono__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).
% fact_less_mono_nat
tff(fact_8321_fact__ge__Suc__0__nat,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).
% fact_ge_Suc_0_nat
tff(fact_8322_dvd__fact,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).
% dvd_fact
tff(fact_8323_fact__diff__Suc,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = 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,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).
% fact_diff_Suc
tff(fact_8324_fact__div__fact__le__pow,axiom,
! [R2: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).
% fact_div_fact_le_pow
tff(fact_8325_sin__coeff__Suc,axiom,
! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).
% sin_coeff_Suc
tff(fact_8326_cos__coeff__Suc,axiom,
! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).
% cos_coeff_Suc
tff(fact_8327_sin__paired,axiom,
! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_kv,Xb)),aa_real_real(sin_real,Xb)) ).
% sin_paired
tff(fact_8328_cos__arcsin,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% cos_arcsin
tff(fact_8329_sin__arccos__abs,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
=> ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(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_8330_sin__arccos,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% sin_arccos
tff(fact_8331_geometric__deriv__sums,axiom,
! [Z: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
=> aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_kw,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_8332_geometric__deriv__sums,axiom,
! [Z: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
=> aa_complex_o(sums_complex(aTP_Lamm_kx(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_8333_powser__sums__zero__iff,axiom,
! [A: fun_nat_complex,Xb: complex] :
( aa_complex_o(sums_complex(aTP_Lamm_fv(A)),Xb)
<=> ( aa_nat_complex(A,zero_zero_nat) = Xb ) ) ).
% powser_sums_zero_iff
tff(fact_8334_powser__sums__zero__iff,axiom,
! [A: fun_nat_real,Xb: real] :
( aa_real_o(sums_real(aTP_Lamm_fx(A)),Xb)
<=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).
% powser_sums_zero_iff
tff(fact_8335_arccos__0,axiom,
aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% arccos_0
tff(fact_8336_arcsin__1,axiom,
aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% arcsin_1
tff(fact_8337_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% arcsin_minus_1
tff(fact_8338_sums__le,axiom,
! [F: fun_nat_nat,G: fun_nat_nat,S2: nat,Ta: nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
=> ( aa_nat_o(sums_nat(F),S2)
=> ( aa_nat_o(sums_nat(G),Ta)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S2),Ta) ) ) ) ).
% sums_le
tff(fact_8339_sums__le,axiom,
! [F: fun_nat_int,G: fun_nat_int,S2: int,Ta: int] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
=> ( aa_int_o(sums_int(F),S2)
=> ( aa_int_o(sums_int(G),Ta)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S2),Ta) ) ) ) ).
% sums_le
tff(fact_8340_sums__le,axiom,
! [F: fun_nat_real,G: fun_nat_real,S2: real,Ta: real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
=> ( aa_real_o(sums_real(F),S2)
=> ( aa_real_o(sums_real(G),Ta)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S2),Ta) ) ) ) ).
% sums_le
tff(fact_8341_sums__mult,axiom,
! [F: fun_nat_real,A: real,C: real] :
( aa_real_o(sums_real(F),A)
=> aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_io(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ) ).
% sums_mult
tff(fact_8342_sums__mult2,axiom,
! [F: fun_nat_real,A: real,C: real] :
( aa_real_o(sums_real(F),A)
=> aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ey(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ).
% sums_mult2
tff(fact_8343_sums__add,axiom,
! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
( aa_real_o(sums_real(F),A)
=> ( aa_real_o(sums_real(G),B)
=> aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fk(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% sums_add
tff(fact_8344_sums__add,axiom,
! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
( aa_nat_o(sums_nat(F),A)
=> ( aa_nat_o(sums_nat(G),B)
=> aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fj(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% sums_add
tff(fact_8345_sums__add,axiom,
! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
( aa_int_o(sums_int(F),A)
=> ( aa_int_o(sums_int(G),B)
=> aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ip(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% sums_add
tff(fact_8346_sums__mult__iff,axiom,
! [C: complex,F: fun_nat_complex,D: complex] :
( ( C != zero_zero_complex )
=> ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_in(C),F)),aa_complex_complex(times_times_complex(C),D))
<=> aa_complex_o(sums_complex(F),D) ) ) ).
% sums_mult_iff
tff(fact_8347_sums__mult__iff,axiom,
! [C: real,F: fun_nat_real,D: real] :
( ( C != zero_zero_real )
=> ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_eu(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))
<=> aa_real_o(sums_real(F),D) ) ) ).
% sums_mult_iff
tff(fact_8348_sums__mult2__iff,axiom,
! [C: complex,F: fun_nat_complex,D: complex] :
( ( C != zero_zero_complex )
=> ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ky(C),F)),aa_complex_complex(times_times_complex(D),C))
<=> aa_complex_o(sums_complex(F),D) ) ) ).
% sums_mult2_iff
tff(fact_8349_sums__mult2__iff,axiom,
! [C: real,F: fun_nat_real,D: real] :
( ( C != zero_zero_real )
=> ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_kz(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C))
<=> aa_real_o(sums_real(F),D) ) ) ).
% sums_mult2_iff
tff(fact_8350_sums__mult__D,axiom,
! [C: complex,F: fun_nat_complex,A: complex] :
( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_in(C),F)),A)
=> ( ( C != zero_zero_complex )
=> aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).
% sums_mult_D
tff(fact_8351_sums__mult__D,axiom,
! [C: real,F: fun_nat_real,A: real] :
( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_eu(C),F)),A)
=> ( ( C != zero_zero_real )
=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).
% sums_mult_D
tff(fact_8352_sums__Suc__imp,axiom,
! [F: fun_nat_complex,S2: complex] :
( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
=> ( aa_complex_o(sums_complex(aTP_Lamm_la(F)),S2)
=> aa_complex_o(sums_complex(F),S2) ) ) ).
% sums_Suc_imp
tff(fact_8353_sums__Suc__imp,axiom,
! [F: fun_nat_real,S2: real] :
( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
=> ( aa_real_o(sums_real(aTP_Lamm_gh(F)),S2)
=> aa_real_o(sums_real(F),S2) ) ) ).
% sums_Suc_imp
tff(fact_8354_sums__Suc__iff,axiom,
! [F: fun_nat_real,S2: real] :
( aa_real_o(sums_real(aTP_Lamm_gh(F)),S2)
<=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_nat_real(F,zero_zero_nat))) ) ).
% sums_Suc_iff
tff(fact_8355_sums__Suc,axiom,
! [F: fun_nat_real,L: real] :
( aa_real_o(sums_real(aTP_Lamm_gh(F)),L)
=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).
% sums_Suc
tff(fact_8356_sums__Suc,axiom,
! [F: fun_nat_nat,L: nat] :
( aa_nat_o(sums_nat(aTP_Lamm_gg(F)),L)
=> aa_nat_o(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_8357_sums__Suc,axiom,
! [F: fun_nat_int,L: int] :
( aa_int_o(sums_int(aTP_Lamm_gr(F)),L)
=> aa_int_o(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_8358_sums__zero__iff__shift,axiom,
! [Nb: nat,F: fun_nat_complex,S2: complex] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
=> ( aa_nat_complex(F,I3) = zero_zero_complex ) )
=> ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lb(Nb),F)),S2)
<=> aa_complex_o(sums_complex(F),S2) ) ) ).
% sums_zero_iff_shift
tff(fact_8359_sums__zero__iff__shift,axiom,
! [Nb: nat,F: fun_nat_real,S2: real] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
=> ( aa_nat_real(F,I3) = zero_zero_real ) )
=> ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_lc(Nb),F)),S2)
<=> aa_real_o(sums_real(F),S2) ) ) ).
% sums_zero_iff_shift
tff(fact_8360_powser__sums__if,axiom,
! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ld(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).
% powser_sums_if
tff(fact_8361_powser__sums__if,axiom,
! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_le(Mb),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)) ).
% powser_sums_if
tff(fact_8362_powser__sums__if,axiom,
! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_lf(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).
% powser_sums_if
tff(fact_8363_powser__sums__zero,axiom,
! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_fv(A)),aa_nat_complex(A,zero_zero_nat)) ).
% powser_sums_zero
tff(fact_8364_powser__sums__zero,axiom,
! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_fx(A)),aa_nat_real(A,zero_zero_nat)) ).
% powser_sums_zero
tff(fact_8365_sums__iff__shift,axiom,
! [F: fun_nat_real,Nb: nat,S2: real] :
( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gj(F),Nb)),S2)
<=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).
% sums_iff_shift
tff(fact_8366_sums__split__initial__segment,axiom,
! [F: fun_nat_real,S2: real,Nb: nat] :
( aa_real_o(sums_real(F),S2)
=> aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gj(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).
% sums_split_initial_segment
tff(fact_8367_sums__iff__shift_H,axiom,
! [F: fun_nat_real,Nb: nat,S2: real] :
( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gj(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
<=> aa_real_o(sums_real(F),S2) ) ).
% sums_iff_shift'
tff(fact_8368_sums__If__finite__set_H,axiom,
! [G: fun_nat_real,S3: real,A3: set_nat,S5: real,F: fun_nat_real] :
( aa_real_o(sums_real(G),S3)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( S5 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lg(G),F),A3)) )
=> aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_lh(G),A3),F)),S5) ) ) ) ).
% sums_If_finite_set'
tff(fact_8369_arccos__less__arccos,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).
% arccos_less_arccos
tff(fact_8370_arccos__less__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).
% arccos_less_mono
tff(fact_8371_arcsin__less__arcsin,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).
% arcsin_less_arcsin
tff(fact_8372_arcsin__less__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).
% arcsin_less_mono
tff(fact_8373_arccos__lt__bounded,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).
% arccos_lt_bounded
tff(fact_8374_sin__arccos__nonzero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) != zero_zero_real ) ) ) ).
% sin_arccos_nonzero
tff(fact_8375_cos__arcsin__nonzero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) != zero_zero_real ) ) ) ).
% cos_arcsin_nonzero
tff(fact_8376_geometric__sums,axiom,
! [C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
=> aa_real_o(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_8377_geometric__sums,axiom,
! [C: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
=> aa_complex_o(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_8378_power__half__series,axiom,
aa_real_o(sums_real(aTP_Lamm_li),one_one_real) ).
% power_half_series
tff(fact_8379_sums__if_H,axiom,
! [G: fun_nat_real,Xb: real] :
( aa_real_o(sums_real(G),Xb)
=> aa_real_o(sums_real(aTP_Lamm_lj(G)),Xb) ) ).
% sums_if'
tff(fact_8380_sums__if,axiom,
! [G: fun_nat_real,Xb: real,F: fun_nat_real,Y: real] :
( aa_real_o(sums_real(G),Xb)
=> ( aa_real_o(sums_real(F),Y)
=> aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_lk(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).
% sums_if
tff(fact_8381_arccos__le__pi2,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).
% arccos_le_pi2
tff(fact_8382_arcsin__lt__bounded,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).
% arcsin_lt_bounded
tff(fact_8383_arcsin__bounded,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).
% arcsin_bounded
tff(fact_8384_arcsin__ubound,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).
% arcsin_ubound
tff(fact_8385_arcsin__lbound,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).
% arcsin_lbound
tff(fact_8386_arcsin__sin,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).
% arcsin_sin
tff(fact_8387_cos__paired,axiom,
! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ll,Xb)),aa_real_real(cos_real,Xb)) ).
% cos_paired
tff(fact_8388_le__arcsin__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xb) ) ) ) ) ) ).
% le_arcsin_iff
tff(fact_8389_arcsin__le__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y)) ) ) ) ) ) ).
% arcsin_le_iff
tff(fact_8390_arcsin__pi,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
& ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).
% arcsin_pi
tff(fact_8391_arcsin,axiom,
! [Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).
% arcsin
tff(fact_8392_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).
% arccos_cos_eq_abs_2pi
tff(fact_8393_diffs__equiv,axiom,
! [C: fun_nat_complex,Xb: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(C),Xb))
=> aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ln(C),Xb)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(C),Xb))) ) ).
% diffs_equiv
tff(fact_8394_diffs__equiv,axiom,
! [C: fun_nat_real,Xb: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_lo(C),Xb))
=> aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_lp(C),Xb)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_lo(C),Xb))) ) ).
% diffs_equiv
tff(fact_8395_monoseq__def,axiom,
! [X6: fun_nat_set_nat] :
( topolo7278393974255667507et_nat(X6)
<=> ( ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M4)),aa_nat_set_nat(X6,N3)) )
| ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N3)),aa_nat_set_nat(X6,M4)) ) ) ) ).
% monoseq_def
tff(fact_8396_monoseq__def,axiom,
! [X6: fun_nat_rat] :
( topolo4267028734544971653eq_rat(X6)
<=> ( ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M4)),aa_nat_rat(X6,N3)) )
| ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,M4)) ) ) ) ).
% monoseq_def
tff(fact_8397_monoseq__def,axiom,
! [X6: fun_nat_num] :
( topolo1459490580787246023eq_num(X6)
<=> ( ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M4)),aa_nat_num(X6,N3)) )
| ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,M4)) ) ) ) ).
% monoseq_def
tff(fact_8398_monoseq__def,axiom,
! [X6: fun_nat_nat] :
( topolo4902158794631467389eq_nat(X6)
<=> ( ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M4)),aa_nat_nat(X6,N3)) )
| ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,M4)) ) ) ) ).
% monoseq_def
tff(fact_8399_monoseq__def,axiom,
! [X6: fun_nat_int] :
( topolo4899668324122417113eq_int(X6)
<=> ( ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M4)),aa_nat_int(X6,N3)) )
| ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,M4)) ) ) ) ).
% monoseq_def
tff(fact_8400_monoseq__def,axiom,
! [X6: fun_nat_real] :
( topolo6980174941875973593q_real(X6)
<=> ( ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N3)) )
| ! [M4: nat,N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,M4)) ) ) ) ).
% monoseq_def
tff(fact_8401_monoI2,axiom,
! [X6: fun_nat_set_nat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,M2)) )
=> topolo7278393974255667507et_nat(X6) ) ).
% monoI2
tff(fact_8402_monoI2,axiom,
! [X6: fun_nat_rat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,M2)) )
=> topolo4267028734544971653eq_rat(X6) ) ).
% monoI2
tff(fact_8403_monoI2,axiom,
! [X6: fun_nat_num] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,M2)) )
=> topolo1459490580787246023eq_num(X6) ) ).
% monoI2
tff(fact_8404_monoI2,axiom,
! [X6: fun_nat_nat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,M2)) )
=> topolo4902158794631467389eq_nat(X6) ) ).
% monoI2
tff(fact_8405_monoI2,axiom,
! [X6: fun_nat_int] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,M2)) )
=> topolo4899668324122417113eq_int(X6) ) ).
% monoI2
tff(fact_8406_monoI2,axiom,
! [X6: fun_nat_real] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,M2)) )
=> topolo6980174941875973593q_real(X6) ) ).
% monoI2
tff(fact_8407_monoI1,axiom,
! [X6: fun_nat_set_nat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M2)),aa_nat_set_nat(X6,N)) )
=> topolo7278393974255667507et_nat(X6) ) ).
% monoI1
tff(fact_8408_monoI1,axiom,
! [X6: fun_nat_rat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M2)),aa_nat_rat(X6,N)) )
=> topolo4267028734544971653eq_rat(X6) ) ).
% monoI1
tff(fact_8409_monoI1,axiom,
! [X6: fun_nat_num] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M2)),aa_nat_num(X6,N)) )
=> topolo1459490580787246023eq_num(X6) ) ).
% monoI1
tff(fact_8410_monoI1,axiom,
! [X6: fun_nat_nat] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M2)),aa_nat_nat(X6,N)) )
=> topolo4902158794631467389eq_nat(X6) ) ).
% monoI1
tff(fact_8411_monoI1,axiom,
! [X6: fun_nat_int] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M2)),aa_nat_int(X6,N)) )
=> topolo4899668324122417113eq_int(X6) ) ).
% monoI1
tff(fact_8412_monoI1,axiom,
! [X6: fun_nat_real] :
( ! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M2)),aa_nat_real(X6,N)) )
=> topolo6980174941875973593q_real(X6) ) ).
% monoI1
tff(fact_8413_pochhammer__double,axiom,
! [Z: complex,Nb: 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))),Nb)) = 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))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),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))))),Nb)) ).
% pochhammer_double
tff(fact_8414_pochhammer__double,axiom,
! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_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))),Nb)) = 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))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Nb)) ).
% pochhammer_double
tff(fact_8415_pochhammer__double,axiom,
! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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))),Nb)) = 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))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).
% pochhammer_double
tff(fact_8416_pochhammer__0,axiom,
! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).
% pochhammer_0
tff(fact_8417_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_8418_pochhammer__0,axiom,
! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).
% pochhammer_0
tff(fact_8419_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_8420_pochhammer__0,axiom,
! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).
% pochhammer_0
tff(fact_8421_pochhammer__pos,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).
% pochhammer_pos
tff(fact_8422_pochhammer__pos,axiom,
! [Xb: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).
% pochhammer_pos
tff(fact_8423_pochhammer__pos,axiom,
! [Xb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).
% pochhammer_pos
tff(fact_8424_pochhammer__pos,axiom,
! [Xb: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).
% pochhammer_pos
tff(fact_8425_pochhammer__neq__0__mono,axiom,
! [A: complex,Mb: nat,Nb: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) != zero_zero_complex )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) != zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_8426_pochhammer__neq__0__mono,axiom,
! [A: real,Mb: nat,Nb: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_8427_pochhammer__neq__0__mono,axiom,
! [A: rat,Mb: nat,Nb: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_8428_pochhammer__eq__0__mono,axiom,
! [A: complex,Nb: nat,Mb: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) = zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_8429_pochhammer__eq__0__mono,axiom,
! [A: real,Nb: nat,Mb: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_8430_pochhammer__eq__0__mono,axiom,
! [A: rat,Nb: nat,Mb: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_8431_pochhammer__fact,axiom,
! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Nb) ).
% pochhammer_fact
tff(fact_8432_pochhammer__fact,axiom,
! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Nb) ).
% pochhammer_fact
tff(fact_8433_pochhammer__fact,axiom,
! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),Nb) ).
% pochhammer_fact
tff(fact_8434_pochhammer__fact,axiom,
! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Nb) ).
% pochhammer_fact
tff(fact_8435_pochhammer__fact,axiom,
! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Nb) ).
% pochhammer_fact
tff(fact_8436_pochhammer__nonneg,axiom,
! [Xb: rat,Nb: nat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).
% pochhammer_nonneg
tff(fact_8437_pochhammer__nonneg,axiom,
! [Xb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).
% pochhammer_nonneg
tff(fact_8438_pochhammer__nonneg,axiom,
! [Xb: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).
% pochhammer_nonneg
tff(fact_8439_pochhammer__nonneg,axiom,
! [Xb: real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).
% pochhammer_nonneg
tff(fact_8440_pochhammer__0__left,axiom,
! [Nb: nat] :
aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).
% pochhammer_0_left
tff(fact_8441_pochhammer__0__left,axiom,
! [Nb: nat] :
aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).
% pochhammer_0_left
tff(fact_8442_pochhammer__0__left,axiom,
! [Nb: nat] :
aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).
% pochhammer_0_left
tff(fact_8443_pochhammer__0__left,axiom,
! [Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).
% pochhammer_0_left
tff(fact_8444_pochhammer__0__left,axiom,
! [Nb: nat] :
aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).
% pochhammer_0_left
tff(fact_8445_pochhammer__rec,axiom,
! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).
% pochhammer_rec
tff(fact_8446_pochhammer__rec,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = 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)),Nb)) ).
% pochhammer_rec
tff(fact_8447_pochhammer__rec,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Nb)) ).
% pochhammer_rec
tff(fact_8448_pochhammer__rec,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = 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)),Nb)) ).
% pochhammer_rec
tff(fact_8449_pochhammer__rec,axiom,
! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = 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)),Nb)) ).
% pochhammer_rec
tff(fact_8450_diffs__def,axiom,
! [C: fun_nat_rat,X3: nat] : aa_nat_rat(diffs_rat(C),X3) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X3))),aa_nat_rat(C,aa_nat_nat(suc,X3))) ).
% diffs_def
tff(fact_8451_diffs__def,axiom,
! [C: fun_nat_int,X3: nat] : aa_nat_int(diffs_int(C),X3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X3))),aa_nat_int(C,aa_nat_nat(suc,X3))) ).
% diffs_def
tff(fact_8452_diffs__def,axiom,
! [C: fun_nat_real,X3: nat] : aa_nat_real(diffs_real(C),X3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X3))),aa_nat_real(C,aa_nat_nat(suc,X3))) ).
% diffs_def
tff(fact_8453_pochhammer__rec_H,axiom,
! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).
% pochhammer_rec'
tff(fact_8454_pochhammer__rec_H,axiom,
! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = 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,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).
% pochhammer_rec'
tff(fact_8455_pochhammer__rec_H,axiom,
! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).
% pochhammer_rec'
tff(fact_8456_pochhammer__rec_H,axiom,
! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = 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,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).
% pochhammer_rec'
tff(fact_8457_pochhammer__Suc,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).
% pochhammer_Suc
tff(fact_8458_pochhammer__Suc,axiom,
! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).
% pochhammer_Suc
tff(fact_8459_pochhammer__Suc,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).
% pochhammer_Suc
tff(fact_8460_pochhammer__Suc,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).
% pochhammer_Suc
tff(fact_8461_pochhammer__of__nat__eq__0__lemma,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
=> ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8462_pochhammer__of__nat__eq__0__lemma,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8463_pochhammer__of__nat__eq__0__lemma,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8464_pochhammer__of__nat__eq__0__lemma,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
=> ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8465_pochhammer__of__nat__eq__0__lemma,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8466_pochhammer__of__nat__eq__0__iff,axiom,
! [Nb: nat,K: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8467_pochhammer__of__nat__eq__0__iff,axiom,
! [Nb: nat,K: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8468_pochhammer__of__nat__eq__0__iff,axiom,
! [Nb: nat,K: nat] :
( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8469_pochhammer__of__nat__eq__0__iff,axiom,
! [Nb: nat,K: nat] :
( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8470_pochhammer__of__nat__eq__0__iff,axiom,
! [Nb: 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,Nb))),K) = zero_zero_real )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8471_pochhammer__eq__0__iff,axiom,
! [A: complex,Nb: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
<=> ? [K2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
& ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_8472_pochhammer__eq__0__iff,axiom,
! [A: rat,Nb: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
<=> ? [K2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
& ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_8473_pochhammer__eq__0__iff,axiom,
! [A: real,Nb: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
<=> ? [K2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
& ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_8474_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) != zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8475_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) != zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8476_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) != zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8477_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) != zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8478_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) != zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8479_pochhammer__product_H,axiom,
! [Z: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).
% pochhammer_product'
tff(fact_8480_pochhammer__product_H,axiom,
! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).
% pochhammer_product'
tff(fact_8481_pochhammer__product_H,axiom,
! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),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,Nb))),Mb)) ).
% pochhammer_product'
tff(fact_8482_pochhammer__product_H,axiom,
! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),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,Nb))),Mb)) ).
% pochhammer_product'
tff(fact_8483_termdiff__converges__all,axiom,
! [C: fun_nat_complex,Xb: complex] :
( ! [X4: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(C),X4))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(C),Xb)) ) ).
% termdiff_converges_all
tff(fact_8484_termdiff__converges__all,axiom,
! [C: fun_nat_real,Xb: real] :
( ! [X4: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(C),X4))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_lo(C),Xb)) ) ).
% termdiff_converges_all
tff(fact_8485_pochhammer__product,axiom,
! [Mb: nat,Nb: nat,Z: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).
% pochhammer_product
tff(fact_8486_pochhammer__product,axiom,
! [Mb: nat,Nb: nat,Z: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).
% pochhammer_product
tff(fact_8487_pochhammer__product,axiom,
! [Mb: nat,Nb: nat,Z: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Mb)),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,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).
% pochhammer_product
tff(fact_8488_pochhammer__product,axiom,
! [Mb: nat,Nb: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),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,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).
% pochhammer_product
tff(fact_8489_pochhammer__absorb__comp,axiom,
! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).
% pochhammer_absorb_comp
tff(fact_8490_pochhammer__absorb__comp,axiom,
! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).
% pochhammer_absorb_comp
tff(fact_8491_pochhammer__absorb__comp,axiom,
! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).
% pochhammer_absorb_comp
tff(fact_8492_pochhammer__absorb__comp,axiom,
! [R2: 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,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).
% pochhammer_absorb_comp
tff(fact_8493_pochhammer__absorb__comp,axiom,
! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),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,R2)),one_one_real)),K)) ).
% pochhammer_absorb_comp
tff(fact_8494_pochhammer__same,axiom,
! [Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).
% pochhammer_same
tff(fact_8495_pochhammer__same,axiom,
! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).
% pochhammer_same
tff(fact_8496_pochhammer__same,axiom,
! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).
% pochhammer_same
tff(fact_8497_pochhammer__same,axiom,
! [Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = 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)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).
% pochhammer_same
tff(fact_8498_pochhammer__same,axiom,
! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = 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)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).
% pochhammer_same
tff(fact_8499_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_8500_pochhammer__minus_H,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_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(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).
% pochhammer_minus'
tff(fact_8501_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_8502_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_8503_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_8504_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_8505_pochhammer__minus,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).
% pochhammer_minus
tff(fact_8506_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_8507_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_8508_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_8509_termdiff__converges,axiom,
! [Xb: real,K5: real,C: fun_nat_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),K5)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X4)),K5)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_iu(C),X4)) )
=> summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_lq(Xb),C)) ) ) ).
% termdiff_converges
tff(fact_8510_termdiff__converges,axiom,
! [Xb: complex,K5: real,C: fun_nat_complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),K5)
=> ( ! [X4: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X4)),K5)
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(C),X4)) )
=> summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(Xb),C)) ) ) ).
% termdiff_converges
tff(fact_8511_fact__double,axiom,
! [Nb: 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))),Nb)) = 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))),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).
% fact_double
tff(fact_8512_fact__double,axiom,
! [Nb: 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))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_num_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))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).
% fact_double
tff(fact_8513_fact__double,axiom,
! [Nb: 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))),Nb)) = 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_num_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))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).
% fact_double
tff(fact_8514_mono__SucI1,axiom,
! [X6: fun_nat_set_nat] :
( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,aa_nat_nat(suc,N)))
=> topolo7278393974255667507et_nat(X6) ) ).
% mono_SucI1
tff(fact_8515_mono__SucI1,axiom,
! [X6: fun_nat_rat] :
( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,aa_nat_nat(suc,N)))
=> topolo4267028734544971653eq_rat(X6) ) ).
% mono_SucI1
tff(fact_8516_mono__SucI1,axiom,
! [X6: fun_nat_num] :
( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,aa_nat_nat(suc,N)))
=> topolo1459490580787246023eq_num(X6) ) ).
% mono_SucI1
tff(fact_8517_mono__SucI1,axiom,
! [X6: fun_nat_nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,aa_nat_nat(suc,N)))
=> topolo4902158794631467389eq_nat(X6) ) ).
% mono_SucI1
tff(fact_8518_mono__SucI1,axiom,
! [X6: fun_nat_int] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,aa_nat_nat(suc,N)))
=> topolo4899668324122417113eq_int(X6) ) ).
% mono_SucI1
tff(fact_8519_mono__SucI1,axiom,
! [X6: fun_nat_real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,aa_nat_nat(suc,N)))
=> topolo6980174941875973593q_real(X6) ) ).
% mono_SucI1
tff(fact_8520_mono__SucI2,axiom,
! [X6: fun_nat_set_nat] :
( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N))),aa_nat_set_nat(X6,N))
=> topolo7278393974255667507et_nat(X6) ) ).
% mono_SucI2
tff(fact_8521_mono__SucI2,axiom,
! [X6: fun_nat_rat] :
( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N))),aa_nat_rat(X6,N))
=> topolo4267028734544971653eq_rat(X6) ) ).
% mono_SucI2
tff(fact_8522_mono__SucI2,axiom,
! [X6: fun_nat_num] :
( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N))),aa_nat_num(X6,N))
=> topolo1459490580787246023eq_num(X6) ) ).
% mono_SucI2
tff(fact_8523_mono__SucI2,axiom,
! [X6: fun_nat_nat] :
( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N))),aa_nat_nat(X6,N))
=> topolo4902158794631467389eq_nat(X6) ) ).
% mono_SucI2
tff(fact_8524_mono__SucI2,axiom,
! [X6: fun_nat_int] :
( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N))),aa_nat_int(X6,N))
=> topolo4899668324122417113eq_int(X6) ) ).
% mono_SucI2
tff(fact_8525_mono__SucI2,axiom,
! [X6: fun_nat_real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N))),aa_nat_real(X6,N))
=> topolo6980174941875973593q_real(X6) ) ).
% mono_SucI2
tff(fact_8526_monoseq__Suc,axiom,
! [X6: fun_nat_set_nat] :
( topolo7278393974255667507et_nat(X6)
<=> ( ! [N3: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N3)),aa_nat_set_nat(X6,aa_nat_nat(suc,N3)))
| ! [N3: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N3))),aa_nat_set_nat(X6,N3)) ) ) ).
% monoseq_Suc
tff(fact_8527_monoseq__Suc,axiom,
! [X6: fun_nat_rat] :
( topolo4267028734544971653eq_rat(X6)
<=> ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,aa_nat_nat(suc,N3)))
| ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N3))),aa_nat_rat(X6,N3)) ) ) ).
% monoseq_Suc
tff(fact_8528_monoseq__Suc,axiom,
! [X6: fun_nat_num] :
( topolo1459490580787246023eq_num(X6)
<=> ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,aa_nat_nat(suc,N3)))
| ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N3))),aa_nat_num(X6,N3)) ) ) ).
% monoseq_Suc
tff(fact_8529_monoseq__Suc,axiom,
! [X6: fun_nat_nat] :
( topolo4902158794631467389eq_nat(X6)
<=> ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,aa_nat_nat(suc,N3)))
| ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N3))),aa_nat_nat(X6,N3)) ) ) ).
% monoseq_Suc
tff(fact_8530_monoseq__Suc,axiom,
! [X6: fun_nat_int] :
( topolo4899668324122417113eq_int(X6)
<=> ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,aa_nat_nat(suc,N3)))
| ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N3))),aa_nat_int(X6,N3)) ) ) ).
% monoseq_Suc
tff(fact_8531_monoseq__Suc,axiom,
! [X6: fun_nat_real] :
( topolo6980174941875973593q_real(X6)
<=> ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,aa_nat_nat(suc,N3)))
| ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N3))),aa_nat_real(X6,N3)) ) ) ).
% monoseq_Suc
tff(fact_8532_pochhammer__times__pochhammer__half,axiom,
! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),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,Nb))) = groups6464643781859351333omplex(aTP_Lamm_ls(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))),Nb)),one_one_nat))) ).
% pochhammer_times_pochhammer_half
tff(fact_8533_pochhammer__times__pochhammer__half,axiom,
! [Z: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_lt(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))),Nb)),one_one_nat))) ).
% pochhammer_times_pochhammer_half
tff(fact_8534_pochhammer__times__pochhammer__half,axiom,
! [Z: real,Nb: 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,Nb))),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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_lu,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))),Nb)),one_one_nat))) ).
% pochhammer_times_pochhammer_half
tff(fact_8535_pochhammer__code,axiom,
! [A: complex,Nb: nat] :
aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_lv(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).
% pochhammer_code
tff(fact_8536_pochhammer__code,axiom,
! [A: rat,Nb: nat] :
aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_lw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).
% pochhammer_code
tff(fact_8537_pochhammer__code,axiom,
! [A: int,Nb: nat] :
aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_lx(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).
% pochhammer_code
tff(fact_8538_pochhammer__code,axiom,
! [A: real,Nb: nat] :
aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ly(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).
% pochhammer_code
tff(fact_8539_pochhammer__code,axiom,
! [A: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_lz(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).
% pochhammer_code
tff(fact_8540_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(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,Nb) )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
& aa_nat_o(aa_nat_fun_nat_o(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,Nb),one_one_nat))) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
tff(fact_8541_of__nat__code,axiom,
! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_ma,Nb),zero_zero_complex) ).
% of_nat_code
tff(fact_8542_of__nat__code,axiom,
! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_mb,Nb),zero_zero_rat) ).
% of_nat_code
tff(fact_8543_of__nat__code,axiom,
! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_mc,Nb),zero_zero_int) ).
% of_nat_code
tff(fact_8544_of__nat__code,axiom,
! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_md),Nb),zero_zero_real) ).
% of_nat_code
tff(fact_8545_of__nat__code,axiom,
! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_me),Nb),zero_zero_nat) ).
% of_nat_code
tff(fact_8546_gchoose__row__sum__weighted,axiom,
! [R2: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_mf(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,Mb))) ).
% gchoose_row_sum_weighted
tff(fact_8547_gchoose__row__sum__weighted,axiom,
! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_mg(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,Mb))) ).
% gchoose_row_sum_weighted
tff(fact_8548_gchoose__row__sum__weighted,axiom,
! [R2: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mh,R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = 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,Mb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,Mb))) ).
% gchoose_row_sum_weighted
tff(fact_8549_prod_Oneutral__const,axiom,
! [A3: set_nat] : groups708209901874060359at_nat(aTP_Lamm_mi,A3) = one_one_nat ).
% prod.neutral_const
tff(fact_8550_prod_Oneutral__const,axiom,
! [A3: set_nat] : groups705719431365010083at_int(aTP_Lamm_mj,A3) = one_one_int ).
% prod.neutral_const
tff(fact_8551_prod_Oneutral__const,axiom,
! [A3: set_int] : groups1705073143266064639nt_int(aTP_Lamm_mk,A3) = one_one_int ).
% prod.neutral_const
tff(fact_8552_prod_Oempty,axiom,
! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).
% prod.empty
tff(fact_8553_prod_Oempty,axiom,
! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).
% prod.empty
tff(fact_8554_prod_Oempty,axiom,
! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).
% prod.empty
tff(fact_8555_prod_Oempty,axiom,
! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).
% prod.empty
tff(fact_8556_prod_Oempty,axiom,
! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).
% prod.empty
tff(fact_8557_prod_Oempty,axiom,
! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).
% prod.empty
tff(fact_8558_prod_Oempty,axiom,
! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).
% prod.empty
tff(fact_8559_prod_Oempty,axiom,
! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).
% prod.empty
tff(fact_8560_prod_Oempty,axiom,
! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).
% prod.empty
tff(fact_8561_prod_Oempty,axiom,
! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).
% prod.empty
tff(fact_8562_prod_Oinfinite,axiom,
! [A3: set_nat,G: fun_nat_complex] :
( ~ aa_set_nat_o(finite_finite_nat,A3)
=> ( groups6464643781859351333omplex(G,A3) = one_one_complex ) ) ).
% prod.infinite
tff(fact_8563_prod_Oinfinite,axiom,
! [A3: set_int,G: fun_int_complex] :
( ~ aa_set_int_o(finite_finite_int,A3)
=> ( groups7440179247065528705omplex(G,A3) = one_one_complex ) ) ).
% prod.infinite
tff(fact_8564_prod_Oinfinite,axiom,
! [A3: set_complex,G: fun_complex_complex] :
( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups3708469109370488835omplex(G,A3) = one_one_complex ) ) ).
% prod.infinite
tff(fact_8565_prod_Oinfinite,axiom,
! [A3: set_nat,G: fun_nat_real] :
( ~ aa_set_nat_o(finite_finite_nat,A3)
=> ( groups129246275422532515t_real(G,A3) = one_one_real ) ) ).
% prod.infinite
tff(fact_8566_prod_Oinfinite,axiom,
! [A3: set_int,G: fun_int_real] :
( ~ aa_set_int_o(finite_finite_int,A3)
=> ( groups2316167850115554303t_real(G,A3) = one_one_real ) ) ).
% prod.infinite
tff(fact_8567_prod_Oinfinite,axiom,
! [A3: set_complex,G: fun_complex_real] :
( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups766887009212190081x_real(G,A3) = one_one_real ) ) ).
% prod.infinite
tff(fact_8568_prod_Oinfinite,axiom,
! [A3: set_nat,G: fun_nat_rat] :
( ~ aa_set_nat_o(finite_finite_nat,A3)
=> ( groups73079841787564623at_rat(G,A3) = one_one_rat ) ) ).
% prod.infinite
tff(fact_8569_prod_Oinfinite,axiom,
! [A3: set_int,G: fun_int_rat] :
( ~ aa_set_int_o(finite_finite_int,A3)
=> ( groups1072433553688619179nt_rat(G,A3) = one_one_rat ) ) ).
% prod.infinite
tff(fact_8570_prod_Oinfinite,axiom,
! [A3: set_complex,G: fun_complex_rat] :
( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups225925009352817453ex_rat(G,A3) = one_one_rat ) ) ).
% prod.infinite
tff(fact_8571_prod_Oinfinite,axiom,
! [A3: set_int,G: fun_int_nat] :
( ~ aa_set_int_o(finite_finite_int,A3)
=> ( groups1707563613775114915nt_nat(G,A3) = one_one_nat ) ) ).
% prod.infinite
tff(fact_8572_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_8573_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_8574_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_8575_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_8576_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_8577_floor__numeral,axiom,
! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).
% floor_numeral
tff(fact_8578_floor__numeral,axiom,
! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).
% floor_numeral
tff(fact_8579_gbinomial__0_I1_J,axiom,
! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).
% gbinomial_0(1)
tff(fact_8580_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_8581_gbinomial__0_I1_J,axiom,
! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).
% gbinomial_0(1)
tff(fact_8582_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_8583_gbinomial__0_I1_J,axiom,
! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).
% gbinomial_0(1)
tff(fact_8584_floor__one,axiom,
aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).
% floor_one
tff(fact_8585_floor__one,axiom,
aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).
% floor_one
tff(fact_8586_prod_Odelta,axiom,
! [S3: set_real,A: real,B: fun_real_complex] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ml(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_complex(B,A),one_one_complex) ) ) ).
% prod.delta
tff(fact_8587_prod_Odelta,axiom,
! [S3: set_nat,A: nat,B: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_mm(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),one_one_complex) ) ) ).
% prod.delta
tff(fact_8588_prod_Odelta,axiom,
! [S3: set_int,A: int,B: fun_int_complex] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_mn(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_complex(B,A),one_one_complex) ) ) ).
% prod.delta
tff(fact_8589_prod_Odelta,axiom,
! [S3: set_complex,A: complex,B: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_mo(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_complex(B,A),one_one_complex) ) ) ).
% prod.delta
tff(fact_8590_prod_Odelta,axiom,
! [S3: set_real,A: real,B: fun_real_real] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_mp(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),one_one_real) ) ) ).
% prod.delta
tff(fact_8591_prod_Odelta,axiom,
! [S3: set_nat,A: nat,B: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mq(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).
% prod.delta
tff(fact_8592_prod_Odelta,axiom,
! [S3: set_int,A: int,B: fun_int_real] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mr(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),one_one_real) ) ) ).
% prod.delta
tff(fact_8593_prod_Odelta,axiom,
! [S3: set_complex,A: complex,B: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ms(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),one_one_real) ) ) ).
% prod.delta
tff(fact_8594_prod_Odelta,axiom,
! [S3: set_real,A: real,B: fun_real_rat] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_mt(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_rat(B,A),one_one_rat) ) ) ).
% prod.delta
tff(fact_8595_prod_Odelta,axiom,
! [S3: set_nat,A: nat,B: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mu(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),one_one_rat) ) ) ).
% prod.delta
tff(fact_8596_prod_Odelta_H,axiom,
! [S3: set_real,A: real,B: fun_real_complex] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_mv(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_complex(B,A),one_one_complex) ) ) ).
% prod.delta'
tff(fact_8597_prod_Odelta_H,axiom,
! [S3: set_nat,A: nat,B: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_mw(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),one_one_complex) ) ) ).
% prod.delta'
tff(fact_8598_prod_Odelta_H,axiom,
! [S3: set_int,A: int,B: fun_int_complex] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_mx(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_complex(B,A),one_one_complex) ) ) ).
% prod.delta'
tff(fact_8599_prod_Odelta_H,axiom,
! [S3: set_complex,A: complex,B: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_my(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_complex(B,A),one_one_complex) ) ) ).
% prod.delta'
tff(fact_8600_prod_Odelta_H,axiom,
! [S3: set_real,A: real,B: fun_real_real] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_mz(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),one_one_real) ) ) ).
% prod.delta'
tff(fact_8601_prod_Odelta_H,axiom,
! [S3: set_nat,A: nat,B: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).
% prod.delta'
tff(fact_8602_prod_Odelta_H,axiom,
! [S3: set_int,A: int,B: fun_int_real] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_nb(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),one_one_real) ) ) ).
% prod.delta'
tff(fact_8603_prod_Odelta_H,axiom,
! [S3: set_complex,A: complex,B: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nc(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),one_one_real) ) ) ).
% prod.delta'
tff(fact_8604_prod_Odelta_H,axiom,
! [S3: set_real,A: real,B: fun_real_rat] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nd(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_rat(B,A),one_one_rat) ) ) ).
% prod.delta'
tff(fact_8605_prod_Odelta_H,axiom,
! [S3: set_nat,A: nat,B: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ne(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),one_one_rat) ) ) ).
% prod.delta'
tff(fact_8606_prod_Oinsert,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xb)),groups2703838992350267259T_real(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8607_prod_Oinsert,axiom,
! [A3: set_real,Xb: real,G: fun_real_real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ~ aa_set_real_o(member_real(Xb),A3)
=> ( groups1681761925125756287l_real(G,insert_real(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xb)),groups1681761925125756287l_real(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8608_prod_Oinsert,axiom,
! [A3: set_nat,Xb: nat,G: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ~ aa_set_nat_o(member_nat(Xb),A3)
=> ( groups129246275422532515t_real(G,insert_nat(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xb)),groups129246275422532515t_real(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8609_prod_Oinsert,axiom,
! [A3: set_int,Xb: int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ~ aa_set_int_o(member_int(Xb),A3)
=> ( groups2316167850115554303t_real(G,insert_int(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xb)),groups2316167850115554303t_real(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8610_prod_Oinsert,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ~ aa_set_complex_o(member_complex(Xb),A3)
=> ( groups766887009212190081x_real(G,insert_complex(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xb)),groups766887009212190081x_real(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8611_prod_Oinsert,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(Xb,A3)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,Xb)),groups5726676334696518183BT_rat(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8612_prod_Oinsert,axiom,
! [A3: set_real,Xb: real,G: fun_real_rat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ~ aa_set_real_o(member_real(Xb),A3)
=> ( groups4061424788464935467al_rat(G,insert_real(Xb,A3)) = aa_rat_rat(times_times_rat(aa_real_rat(G,Xb)),groups4061424788464935467al_rat(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8613_prod_Oinsert,axiom,
! [A3: set_nat,Xb: nat,G: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ~ aa_set_nat_o(member_nat(Xb),A3)
=> ( groups73079841787564623at_rat(G,insert_nat(Xb,A3)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Xb)),groups73079841787564623at_rat(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8614_prod_Oinsert,axiom,
! [A3: set_int,Xb: int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ~ aa_set_int_o(member_int(Xb),A3)
=> ( groups1072433553688619179nt_rat(G,insert_int(Xb,A3)) = aa_rat_rat(times_times_rat(aa_int_rat(G,Xb)),groups1072433553688619179nt_rat(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8615_prod_Oinsert,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ~ aa_set_complex_o(member_complex(Xb),A3)
=> ( groups225925009352817453ex_rat(G,insert_complex(Xb,A3)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,Xb)),groups225925009352817453ex_rat(G,A3)) ) ) ) ).
% prod.insert
tff(fact_8616_prod_OlessThan__Suc,axiom,
! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).
% prod.lessThan_Suc
tff(fact_8617_prod_OlessThan__Suc,axiom,
! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).
% prod.lessThan_Suc
tff(fact_8618_prod_OlessThan__Suc,axiom,
! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).
% prod.lessThan_Suc
tff(fact_8619_prod_OlessThan__Suc,axiom,
! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).
% prod.lessThan_Suc
tff(fact_8620_zero__le__floor,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).
% zero_le_floor
tff(fact_8621_zero__le__floor,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb) ) ).
% zero_le_floor
tff(fact_8622_floor__less__zero,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).
% floor_less_zero
tff(fact_8623_floor__less__zero,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),zero_zero_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat) ) ).
% floor_less_zero
tff(fact_8624_numeral__le__floor,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),Xb) ) ).
% numeral_le_floor
tff(fact_8625_numeral__le__floor,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),Xb) ) ).
% numeral_le_floor
tff(fact_8626_zero__less__floor,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).
% zero_less_floor
tff(fact_8627_zero__less__floor,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).
% zero_less_floor
tff(fact_8628_floor__le__zero,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).
% floor_le_zero
tff(fact_8629_floor__le__zero,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),zero_zero_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).
% floor_le_zero
tff(fact_8630_floor__less__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,V)) ) ).
% floor_less_numeral
tff(fact_8631_floor__less__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_num_rat(numeral_numeral_rat,V)) ) ).
% floor_less_numeral
tff(fact_8632_one__le__floor,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).
% one_le_floor
tff(fact_8633_one__le__floor,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).
% one_le_floor
tff(fact_8634_floor__less__one,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).
% floor_less_one
tff(fact_8635_floor__less__one,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).
% floor_less_one
tff(fact_8636_floor__neg__numeral,axiom,
! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% floor_neg_numeral
tff(fact_8637_floor__neg__numeral,axiom,
! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% floor_neg_numeral
tff(fact_8638_floor__diff__numeral,axiom,
! [Xb: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).
% floor_diff_numeral
tff(fact_8639_floor__diff__numeral,axiom,
! [Xb: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).
% floor_diff_numeral
tff(fact_8640_floor__diff__one,axiom,
! [Xb: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) ).
% floor_diff_one
tff(fact_8641_floor__diff__one,axiom,
! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int) ).
% floor_diff_one
tff(fact_8642_floor__numeral__power,axiom,
! [Xb: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).
% floor_numeral_power
tff(fact_8643_floor__numeral__power,axiom,
! [Xb: num,Nb: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).
% floor_numeral_power
tff(fact_8644_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_complex,Mb: nat,Nb: nat] :
groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).
% prod.cl_ivl_Suc
tff(fact_8645_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_real,Mb: nat,Nb: nat] :
groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).
% prod.cl_ivl_Suc
tff(fact_8646_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_rat,Mb: nat,Nb: nat] :
groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).
% prod.cl_ivl_Suc
tff(fact_8647_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_nat,Mb: nat,Nb: nat] :
groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).
% prod.cl_ivl_Suc
tff(fact_8648_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_int,Mb: nat,Nb: nat] :
groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).
% prod.cl_ivl_Suc
tff(fact_8649_numeral__less__floor,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xb) ) ).
% numeral_less_floor
tff(fact_8650_numeral__less__floor,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xb) ) ).
% numeral_less_floor
tff(fact_8651_floor__le__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).
% floor_le_numeral
tff(fact_8652_floor__le__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_num_int(numeral_numeral_int,V))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).
% floor_le_numeral
tff(fact_8653_one__less__floor,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb) ) ).
% one_less_floor
tff(fact_8654_one__less__floor,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb) ) ).
% one_less_floor
tff(fact_8655_floor__le__one,axiom,
! [Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).
% floor_le_one
tff(fact_8656_floor__le__one,axiom,
! [Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).
% floor_le_one
tff(fact_8657_neg__numeral__le__floor,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xb) ) ).
% neg_numeral_le_floor
tff(fact_8658_neg__numeral__le__floor,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xb) ) ).
% neg_numeral_le_floor
tff(fact_8659_floor__less__neg__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).
% floor_less_neg_numeral
tff(fact_8660_floor__less__neg__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).
% floor_less_neg_numeral
tff(fact_8661_neg__numeral__less__floor,axiom,
! [V: num,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xb) ) ).
% neg_numeral_less_floor
tff(fact_8662_neg__numeral__less__floor,axiom,
! [V: num,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xb) ) ).
% neg_numeral_less_floor
tff(fact_8663_floor__le__neg__numeral,axiom,
! [Xb: real,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).
% floor_le_neg_numeral
tff(fact_8664_floor__le__neg__numeral,axiom,
! [Xb: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).
% floor_le_neg_numeral
tff(fact_8665_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_complex_complex,A3: set_complex] :
( ( groups3708469109370488835omplex(G,A3) != one_one_complex )
=> ~ ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),A3)
=> ( aa_complex_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8666_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_real_complex,A3: set_real] :
( ( groups713298508707869441omplex(G,A3) != one_one_complex )
=> ~ ! [A4: real] :
( aa_set_real_o(member_real(A4),A3)
=> ( aa_real_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8667_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_nat_complex,A3: set_nat] :
( ( groups6464643781859351333omplex(G,A3) != one_one_complex )
=> ~ ! [A4: nat] :
( aa_set_nat_o(member_nat(A4),A3)
=> ( aa_nat_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8668_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_int_complex,A3: set_int] :
( ( groups7440179247065528705omplex(G,A3) != one_one_complex )
=> ~ ! [A4: int] :
( aa_set_int_o(member_int(A4),A3)
=> ( aa_int_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8669_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_complex_real,A3: set_complex] :
( ( groups766887009212190081x_real(G,A3) != one_one_real )
=> ~ ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),A3)
=> ( aa_complex_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8670_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_real_real,A3: set_real] :
( ( groups1681761925125756287l_real(G,A3) != one_one_real )
=> ~ ! [A4: real] :
( aa_set_real_o(member_real(A4),A3)
=> ( aa_real_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8671_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_nat_real,A3: set_nat] :
( ( groups129246275422532515t_real(G,A3) != one_one_real )
=> ~ ! [A4: nat] :
( aa_set_nat_o(member_nat(A4),A3)
=> ( aa_nat_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8672_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_int_real,A3: set_int] :
( ( groups2316167850115554303t_real(G,A3) != one_one_real )
=> ~ ! [A4: int] :
( aa_set_int_o(member_int(A4),A3)
=> ( aa_int_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8673_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_complex_rat,A3: set_complex] :
( ( groups225925009352817453ex_rat(G,A3) != one_one_rat )
=> ~ ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),A3)
=> ( aa_complex_rat(G,A4) = one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8674_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_real_rat,A3: set_real] :
( ( groups4061424788464935467al_rat(G,A3) != one_one_rat )
=> ~ ! [A4: real] :
( aa_set_real_o(member_real(A4),A3)
=> ( aa_real_rat(G,A4) = one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8675_prod_Oneutral,axiom,
! [A3: set_nat,G: fun_nat_nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> ( aa_nat_nat(G,X4) = one_one_nat ) )
=> ( groups708209901874060359at_nat(G,A3) = one_one_nat ) ) ).
% prod.neutral
tff(fact_8676_prod_Oneutral,axiom,
! [A3: set_nat,G: fun_nat_int] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> ( aa_nat_int(G,X4) = one_one_int ) )
=> ( groups705719431365010083at_int(G,A3) = one_one_int ) ) ).
% prod.neutral
tff(fact_8677_prod_Oneutral,axiom,
! [A3: set_int,G: fun_int_int] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> ( aa_int_int(G,X4) = one_one_int ) )
=> ( groups1705073143266064639nt_int(G,A3) = one_one_int ) ) ).
% prod.neutral
tff(fact_8678_prod_Odistrib,axiom,
! [G: fun_nat_nat,H: fun_nat_nat,A3: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nf(G),H),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A3)),groups708209901874060359at_nat(H,A3)) ).
% prod.distrib
tff(fact_8679_prod_Odistrib,axiom,
! [G: fun_nat_int,H: fun_nat_int,A3: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ng(G),H),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A3)),groups705719431365010083at_int(H,A3)) ).
% prod.distrib
tff(fact_8680_prod_Odistrib,axiom,
! [G: fun_int_int,H: fun_int_int,A3: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_nh(G),H),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A3)),groups1705073143266064639nt_int(H,A3)) ).
% prod.distrib
tff(fact_8681_prod__power__distrib,axiom,
! [F: fun_nat_nat,A3: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A3)),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ni(F),Nb),A3) ).
% prod_power_distrib
tff(fact_8682_prod__power__distrib,axiom,
! [F: fun_nat_int,A3: set_nat,Nb: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A3)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_nj(F),Nb),A3) ).
% prod_power_distrib
tff(fact_8683_prod__power__distrib,axiom,
! [F: fun_int_int,A3: set_int,Nb: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A3)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_nk(F),Nb),A3) ).
% prod_power_distrib
tff(fact_8684_prod__mono,axiom,
! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(G,A3)) ) ).
% prod_mono
tff(fact_8685_prod__mono,axiom,
! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(G,A3)) ) ).
% prod_mono
tff(fact_8686_prod__mono,axiom,
! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(G,A3)) ) ).
% prod_mono
tff(fact_8687_prod__mono,axiom,
! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
( ! [I3: int] :
( aa_set_int_o(member_int(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ).
% prod_mono
tff(fact_8688_prod__mono,axiom,
! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A3)),groups861055069439313189ex_nat(G,A3)) ) ).
% prod_mono
tff(fact_8689_prod__mono,axiom,
! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A3)),groups4696554848551431203al_nat(G,A3)) ) ).
% prod_mono
tff(fact_8690_prod__mono,axiom,
! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
( ! [I3: int] :
( aa_set_int_o(member_int(I3),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,A3)) ) ).
% prod_mono
tff(fact_8691_prod__mono,axiom,
! [A3: set_complex,F: fun_complex_int,G: fun_complex_int] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I3))
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(G,A3)) ) ).
% prod_mono
tff(fact_8692_prod__mono,axiom,
! [A3: set_real,F: fun_real_int,G: fun_real_int] :
( ! [I3: real] :
( aa_set_real_o(member_real(I3),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3))
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(G,A3)) ) ).
% prod_mono
tff(fact_8693_prod__mono,axiom,
! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I3)),aa_complex_real(G,I3)) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(G,A3)) ) ).
% prod_mono
tff(fact_8694_prod__nonneg,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups708209901874060359at_nat(F,A3)) ) ).
% prod_nonneg
tff(fact_8695_prod__nonneg,axiom,
! [A3: set_nat,F: fun_nat_int] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).
% prod_nonneg
tff(fact_8696_prod__nonneg,axiom,
! [A3: set_int,F: fun_int_int] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).
% prod_nonneg
tff(fact_8697_prod__pos,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A3)) ) ).
% prod_pos
tff(fact_8698_prod__pos,axiom,
! [A3: set_nat,F: fun_nat_int] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).
% prod_pos
tff(fact_8699_prod__pos,axiom,
! [A3: set_int,F: fun_int_int] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).
% prod_pos
tff(fact_8700_prod__ge__1,axiom,
! [A3: set_complex,F: fun_complex_rat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A3)) ) ).
% prod_ge_1
tff(fact_8701_prod__ge__1,axiom,
! [A3: set_real,F: fun_real_rat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A3)) ) ).
% prod_ge_1
tff(fact_8702_prod__ge__1,axiom,
! [A3: set_nat,F: fun_nat_rat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A3)) ) ).
% prod_ge_1
tff(fact_8703_prod__ge__1,axiom,
! [A3: set_int,F: fun_int_rat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3)) ) ).
% prod_ge_1
tff(fact_8704_prod__ge__1,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_complex_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups861055069439313189ex_nat(F,A3)) ) ).
% prod_ge_1
tff(fact_8705_prod__ge__1,axiom,
! [A3: set_real,F: fun_real_nat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A3)) ) ).
% prod_ge_1
tff(fact_8706_prod__ge__1,axiom,
! [A3: set_int,F: fun_int_nat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A3)) ) ).
% prod_ge_1
tff(fact_8707_prod__ge__1,axiom,
! [A3: set_complex,F: fun_complex_int] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups858564598930262913ex_int(F,A3)) ) ).
% prod_ge_1
tff(fact_8708_prod__ge__1,axiom,
! [A3: set_real,F: fun_real_int] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,X4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A3)) ) ).
% prod_ge_1
tff(fact_8709_prod__ge__1,axiom,
! [A3: set_complex,F: fun_complex_real] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A3)) ) ).
% prod_ge_1
tff(fact_8710_floor__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).
% floor_mono
tff(fact_8711_floor__mono,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ).
% floor_mono
tff(fact_8712_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_nl(F),A,B),one_one_complex) ).
% prod_atLeastAtMost_code
tff(fact_8713_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_real,A: nat,B: nat] : groups129246275422532515t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_nm(F),A),B),one_one_real) ).
% prod_atLeastAtMost_code
tff(fact_8714_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_nn(F),A,B),one_one_rat) ).
% prod_atLeastAtMost_code
tff(fact_8715_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_nat,A: nat,B: nat] : groups708209901874060359at_nat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_no(F),A),B),one_one_nat) ).
% prod_atLeastAtMost_code
tff(fact_8716_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_np(F),A,B),one_one_int) ).
% prod_atLeastAtMost_code
tff(fact_8717_of__int__floor__le,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))),Xb) ).
% of_int_floor_le
tff(fact_8718_of__int__floor__le,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb))),Xb) ).
% of_int_floor_le
tff(fact_8719_floor__less__cancel,axiom,
! [Xb: real,Y: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% floor_less_cancel
tff(fact_8720_floor__less__cancel,axiom,
! [Xb: rat,Y: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).
% floor_less_cancel
tff(fact_8721_prod_Ointer__filter,axiom,
! [A3: set_real,G: fun_real_complex,P: fun_real_o] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups713298508707869441omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_nq(A3),P))) = groups713298508707869441omplex(aa_fun1098856492951973088omplex(aTP_Lamm_nr(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8722_prod_Ointer__filter,axiom,
! [A3: set_nat,G: fun_nat_complex,P: fun_nat_o] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups6464643781859351333omplex(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ns(A3),P))) = groups6464643781859351333omplex(aa_fun1400689748890728928omplex(aTP_Lamm_nt(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8723_prod_Ointer__filter,axiom,
! [A3: set_int,G: fun_int_complex,P: fun_int_o] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups7440179247065528705omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_nu(A3),P))) = groups7440179247065528705omplex(aa_fun6439487526662521568omplex(aTP_Lamm_nv(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8724_prod_Ointer__filter,axiom,
! [A3: set_complex,G: fun_complex_complex,P: fun_complex_o] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups3708469109370488835omplex(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_nw(A3),P))) = groups3708469109370488835omplex(aa_fun3681620399087435360omplex(aTP_Lamm_nx(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8725_prod_Ointer__filter,axiom,
! [A3: set_real,G: fun_real_real,P: fun_real_o] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups1681761925125756287l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_nq(A3),P))) = groups1681761925125756287l_real(aa_fun4448911596306054750l_real(aTP_Lamm_ny(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8726_prod_Ointer__filter,axiom,
! [A3: set_nat,G: fun_nat_real,P: fun_nat_o] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups129246275422532515t_real(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ns(A3),P))) = groups129246275422532515t_real(aa_fun5510454223943551838t_real(aTP_Lamm_nz(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8727_prod_Ointer__filter,axiom,
! [A3: set_int,G: fun_int_real,P: fun_int_o] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups2316167850115554303t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_nu(A3),P))) = groups2316167850115554303t_real(aa_fun7506669883404309598t_real(aTP_Lamm_oa(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8728_prod_Ointer__filter,axiom,
! [A3: set_complex,G: fun_complex_real,P: fun_complex_o] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups766887009212190081x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_nw(A3),P))) = groups766887009212190081x_real(aa_fun8261057609414086622x_real(aTP_Lamm_ob(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8729_prod_Ointer__filter,axiom,
! [A3: set_real,G: fun_real_rat,P: fun_real_o] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups4061424788464935467al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_nq(A3),P))) = groups4061424788464935467al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_oc(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8730_prod_Ointer__filter,axiom,
! [A3: set_nat,G: fun_nat_rat,P: fun_nat_o] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups73079841787564623at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ns(A3),P))) = groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_od(G),P),A3) ) ) ).
% prod.inter_filter
tff(fact_8731_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: fun_nat_nat,Mb: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups708209901874060359at_nat(aTP_Lamm_gg(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.shift_bounds_cl_Suc_ivl
tff(fact_8732_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_gr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.shift_bounds_cl_Suc_ivl
tff(fact_8733_power__sum,axiom,
! [C: real,F: fun_nat_nat,A3: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_oe(C),F),A3) ).
% power_sum
tff(fact_8734_power__sum,axiom,
! [C: complex,F: fun_nat_nat,A3: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_of(C),F),A3) ).
% power_sum
tff(fact_8735_power__sum,axiom,
! [C: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_og(C),F),A3) ).
% power_sum
tff(fact_8736_power__sum,axiom,
! [C: int,F: fun_nat_nat,A3: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_oh(C),F),A3) ).
% power_sum
tff(fact_8737_power__sum,axiom,
! [C: int,F: fun_int_nat,A3: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A3)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_oi(C),F),A3) ).
% power_sum
tff(fact_8738_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gi(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.shift_bounds_cl_nat_ivl
tff(fact_8739_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: fun_nat_int,Mb: nat,K: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_oj(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.shift_bounds_cl_nat_ivl
tff(fact_8740_prod__le__1,axiom,
! [A3: set_complex,F: fun_complex_rat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),one_one_rat) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),one_one_rat) ) ).
% prod_le_1
tff(fact_8741_prod__le__1,axiom,
! [A3: set_real,F: fun_real_rat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X4)),one_one_rat) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),one_one_rat) ) ).
% prod_le_1
tff(fact_8742_prod__le__1,axiom,
! [A3: set_nat,F: fun_nat_rat] :
( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),one_one_rat) ) ).
% prod_le_1
tff(fact_8743_prod__le__1,axiom,
! [A3: set_int,F: fun_int_rat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat) ) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),one_one_rat) ) ).
% prod_le_1
tff(fact_8744_prod__le__1,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),one_one_nat) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A3)),one_one_nat) ) ).
% prod_le_1
tff(fact_8745_prod__le__1,axiom,
! [A3: set_real,F: fun_real_nat] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A3)),one_one_nat) ) ).
% prod_le_1
tff(fact_8746_prod__le__1,axiom,
! [A3: set_int,F: fun_int_nat] :
( ! [X4: int] :
( aa_set_int_o(member_int(X4),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),one_one_nat) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),one_one_nat) ) ).
% prod_le_1
tff(fact_8747_prod__le__1,axiom,
! [A3: set_complex,F: fun_complex_int] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4))
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),one_one_int) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A3)),one_one_int) ) ).
% prod_le_1
tff(fact_8748_prod__le__1,axiom,
! [A3: set_real,F: fun_real_int] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),one_one_int) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),one_one_int) ) ).
% prod_le_1
tff(fact_8749_prod__le__1,axiom,
! [A3: set_complex,F: fun_complex_real] :
( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),one_one_real) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),one_one_real) ) ).
% prod_le_1
tff(fact_8750_prod_Orelated,axiom,
! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
& aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
=> ( aa_set_nat_o(finite_finite_nat,S3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4)) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,groups6464643781859351333omplex(H,S3)),groups6464643781859351333omplex(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8751_prod_Orelated,axiom,
! [R: fun_co1148497455491383662plex_o,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
& aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X4)),aa_int_complex(G,X4)) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,groups7440179247065528705omplex(H,S3)),groups7440179247065528705omplex(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8752_prod_Orelated,axiom,
! [R: fun_co1148497455491383662plex_o,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
& aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(H,X4)),aa_complex_complex(G,X4)) )
=> aa_complex_o(aa_com5808801942499186519plex_o(R,groups3708469109370488835omplex(H,S3)),groups3708469109370488835omplex(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8753_prod_Orelated,axiom,
! [R: fun_real_fun_real_o,S3: set_nat,H: fun_nat_real,G: fun_nat_real] :
( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
& aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
=> aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
=> ( aa_set_nat_o(finite_finite_nat,S3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
=> aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X4)),aa_nat_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(R,groups129246275422532515t_real(H,S3)),groups129246275422532515t_real(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8754_prod_Orelated,axiom,
! [R: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
& aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
=> aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X4)),aa_int_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(R,groups2316167850115554303t_real(H,S3)),groups2316167850115554303t_real(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8755_prod_Orelated,axiom,
! [R: fun_real_fun_real_o,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
& aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
=> aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X4)),aa_complex_real(G,X4)) )
=> aa_real_o(aa_real_fun_real_o(R,groups766887009212190081x_real(H,S3)),groups766887009212190081x_real(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8756_prod_Orelated,axiom,
! [R: fun_rat_fun_rat_o,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
& aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23)) )
=> ( aa_set_nat_o(finite_finite_nat,S3)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),S3)
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(H,S3)),groups73079841787564623at_rat(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8757_prod_Orelated,axiom,
! [R: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
& aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X4)),aa_int_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(R,groups1072433553688619179nt_rat(H,S3)),groups1072433553688619179nt_rat(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8758_prod_Orelated,axiom,
! [R: fun_rat_fun_rat_o,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
& aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23)) )
=> ( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4)) )
=> aa_rat_o(aa_rat_fun_rat_o(R,groups225925009352817453ex_rat(H,S3)),groups225925009352817453ex_rat(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8759_prod_Orelated,axiom,
! [R: fun_nat_fun_nat_o,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
=> ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
& aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
=> aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y23)) )
=> ( aa_set_int_o(finite_finite_int,S3)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X4)),aa_int_nat(G,X4)) )
=> aa_nat_o(aa_nat_fun_nat_o(R,groups1707563613775114915nt_nat(H,S3)),groups1707563613775114915nt_nat(G,S3)) ) ) ) ) ).
% prod.related
tff(fact_8760_prod_Oinsert__if,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xb,A3)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),groups2703838992350267259T_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xb)),groups2703838992350267259T_real(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8761_prod_Oinsert__if,axiom,
! [A3: set_real,G: fun_real_real,Xb: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups1681761925125756287l_real(G,insert_real(Xb,A3)) = $ite(aa_set_real_o(member_real(Xb),A3),groups1681761925125756287l_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xb)),groups1681761925125756287l_real(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8762_prod_Oinsert__if,axiom,
! [A3: set_nat,G: fun_nat_real,Xb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups129246275422532515t_real(G,insert_nat(Xb,A3)) = $ite(aa_set_nat_o(member_nat(Xb),A3),groups129246275422532515t_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xb)),groups129246275422532515t_real(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8763_prod_Oinsert__if,axiom,
! [A3: set_int,G: fun_int_real,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups2316167850115554303t_real(G,insert_int(Xb,A3)) = $ite(aa_set_int_o(member_int(Xb),A3),groups2316167850115554303t_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xb)),groups2316167850115554303t_real(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8764_prod_Oinsert__if,axiom,
! [A3: set_complex,G: fun_complex_real,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups766887009212190081x_real(G,insert_complex(Xb,A3)) = $ite(aa_set_complex_o(member_complex(Xb),A3),groups766887009212190081x_real(G,A3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xb)),groups766887009212190081x_real(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8765_prod_Oinsert__if,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(Xb,A3)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),groups5726676334696518183BT_rat(G,A3),aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,Xb)),groups5726676334696518183BT_rat(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8766_prod_Oinsert__if,axiom,
! [A3: set_real,G: fun_real_rat,Xb: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups4061424788464935467al_rat(G,insert_real(Xb,A3)) = $ite(aa_set_real_o(member_real(Xb),A3),groups4061424788464935467al_rat(G,A3),aa_rat_rat(times_times_rat(aa_real_rat(G,Xb)),groups4061424788464935467al_rat(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8767_prod_Oinsert__if,axiom,
! [A3: set_nat,G: fun_nat_rat,Xb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups73079841787564623at_rat(G,insert_nat(Xb,A3)) = $ite(aa_set_nat_o(member_nat(Xb),A3),groups73079841787564623at_rat(G,A3),aa_rat_rat(times_times_rat(aa_nat_rat(G,Xb)),groups73079841787564623at_rat(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8768_prod_Oinsert__if,axiom,
! [A3: set_int,G: fun_int_rat,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups1072433553688619179nt_rat(G,insert_int(Xb,A3)) = $ite(aa_set_int_o(member_int(Xb),A3),groups1072433553688619179nt_rat(G,A3),aa_rat_rat(times_times_rat(aa_int_rat(G,Xb)),groups1072433553688619179nt_rat(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8769_prod_Oinsert__if,axiom,
! [A3: set_complex,G: fun_complex_rat,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups225925009352817453ex_rat(G,insert_complex(Xb,A3)) = $ite(aa_set_complex_o(member_complex(Xb),A3),groups225925009352817453ex_rat(G,A3),aa_rat_rat(times_times_rat(aa_complex_rat(G,Xb)),groups225925009352817453ex_rat(G,A3))) ) ) ).
% prod.insert_if
tff(fact_8770_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_complex,H: fun_real_complex] :
( aa_set_real_o(finite_finite_real,S5)
=> ( aa_set_real_o(finite_finite_real,T3)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> aa_set_real_o(member_real(aa_real_real(J,A4)),minus_minus_set_real(T4,T3)) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> aa_set_real_o(member_real(aa_real_real(I,B4)),minus_minus_set_real(S3,S5)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S5)
=> ( aa_real_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),T3)
=> ( aa_real_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S3)
=> ( aa_real_complex(H,aa_real_real(J,A4)) = aa_real_complex(G,A4) ) )
=> ( groups713298508707869441omplex(G,S3) = groups713298508707869441omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8771_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T3: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_complex,H: fun_int_complex] :
( aa_set_real_o(finite_finite_real,S5)
=> ( aa_set_int_o(finite_finite_int,T3)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> ( aa_int_real(I,aa_real_int(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> aa_set_int_o(member_int(aa_real_int(J,A4)),minus_minus_set_int(T4,T3)) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(T4,T3))
=> ( aa_real_int(J,aa_int_real(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(T4,T3))
=> aa_set_real_o(member_real(aa_int_real(I,B4)),minus_minus_set_real(S3,S5)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S5)
=> ( aa_real_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),T3)
=> ( aa_int_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S3)
=> ( aa_int_complex(H,aa_real_int(J,A4)) = aa_real_complex(G,A4) ) )
=> ( groups713298508707869441omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8772_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T3: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_complex,H: fun_complex_complex] :
( aa_set_real_o(finite_finite_real,S5)
=> ( aa_set_complex_o(finite3207457112153483333omplex,T3)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> ( aa_complex_real(I,aa_real_complex(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> aa_set_complex_o(member_complex(aa_real_complex(J,A4)),minus_811609699411566653omplex(T4,T3)) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T4,T3))
=> ( aa_real_complex(J,aa_complex_real(I,B4)) = B4 ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T4,T3))
=> aa_set_real_o(member_real(aa_complex_real(I,B4)),minus_minus_set_real(S3,S5)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S5)
=> ( aa_real_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),T3)
=> ( aa_complex_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S3)
=> ( aa_complex_complex(H,aa_real_complex(J,A4)) = aa_real_complex(G,A4) ) )
=> ( groups713298508707869441omplex(G,S3) = groups3708469109370488835omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8773_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T3: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T4: set_real,G: fun_int_complex,H: fun_real_complex] :
( aa_set_int_o(finite_finite_int,S5)
=> ( aa_set_real_o(finite_finite_real,T3)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(S3,S5))
=> ( aa_real_int(I,aa_int_real(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(S3,S5))
=> aa_set_real_o(member_real(aa_int_real(J,A4)),minus_minus_set_real(T4,T3)) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> ( aa_int_real(J,aa_real_int(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> aa_set_int_o(member_int(aa_real_int(I,B4)),minus_minus_set_int(S3,S5)) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),S5)
=> ( aa_int_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),T3)
=> ( aa_real_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),S3)
=> ( aa_real_complex(H,aa_int_real(J,A4)) = aa_int_complex(G,A4) ) )
=> ( groups7440179247065528705omplex(G,S3) = groups713298508707869441omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8774_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T3: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T4: set_int,G: fun_int_complex,H: fun_int_complex] :
( aa_set_int_o(finite_finite_int,S5)
=> ( aa_set_int_o(finite_finite_int,T3)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(S3,S5))
=> ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(S3,S5))
=> aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T4,T3)) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(T4,T3))
=> ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(T4,T3))
=> aa_set_int_o(member_int(aa_int_int(I,B4)),minus_minus_set_int(S3,S5)) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),S5)
=> ( aa_int_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),T3)
=> ( aa_int_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),S3)
=> ( aa_int_complex(H,aa_int_int(J,A4)) = aa_int_complex(G,A4) ) )
=> ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8775_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T3: set_complex,S3: set_int,I: fun_complex_int,J: fun_int_complex,T4: set_complex,G: fun_int_complex,H: fun_complex_complex] :
( aa_set_int_o(finite_finite_int,S5)
=> ( aa_set_complex_o(finite3207457112153483333omplex,T3)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(S3,S5))
=> ( aa_complex_int(I,aa_int_complex(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(S3,S5))
=> aa_set_complex_o(member_complex(aa_int_complex(J,A4)),minus_811609699411566653omplex(T4,T3)) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T4,T3))
=> ( aa_int_complex(J,aa_complex_int(I,B4)) = B4 ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T4,T3))
=> aa_set_int_o(member_int(aa_complex_int(I,B4)),minus_minus_set_int(S3,S5)) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),S5)
=> ( aa_int_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),T3)
=> ( aa_complex_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),S3)
=> ( aa_complex_complex(H,aa_int_complex(J,A4)) = aa_int_complex(G,A4) ) )
=> ( groups7440179247065528705omplex(G,S3) = groups3708469109370488835omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8776_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T3: set_real,S3: set_complex,I: fun_real_complex,J: fun_complex_real,T4: set_real,G: fun_complex_complex,H: fun_real_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S5)
=> ( aa_set_real_o(finite_finite_real,T3)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(S3,S5))
=> ( aa_real_complex(I,aa_complex_real(J,A4)) = A4 ) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(S3,S5))
=> aa_set_real_o(member_real(aa_complex_real(J,A4)),minus_minus_set_real(T4,T3)) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> ( aa_complex_real(J,aa_real_complex(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> aa_set_complex_o(member_complex(aa_real_complex(I,B4)),minus_811609699411566653omplex(S3,S5)) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),S5)
=> ( aa_complex_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),T3)
=> ( aa_real_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),S3)
=> ( aa_real_complex(H,aa_complex_real(J,A4)) = aa_complex_complex(G,A4) ) )
=> ( groups3708469109370488835omplex(G,S3) = groups713298508707869441omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8777_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T3: set_int,S3: set_complex,I: fun_int_complex,J: fun_complex_int,T4: set_int,G: fun_complex_complex,H: fun_int_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S5)
=> ( aa_set_int_o(finite_finite_int,T3)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(S3,S5))
=> ( aa_int_complex(I,aa_complex_int(J,A4)) = A4 ) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(S3,S5))
=> aa_set_int_o(member_int(aa_complex_int(J,A4)),minus_minus_set_int(T4,T3)) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(T4,T3))
=> ( aa_complex_int(J,aa_int_complex(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(T4,T3))
=> aa_set_complex_o(member_complex(aa_int_complex(I,B4)),minus_811609699411566653omplex(S3,S5)) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),S5)
=> ( aa_complex_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),T3)
=> ( aa_int_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),S3)
=> ( aa_int_complex(H,aa_complex_int(J,A4)) = aa_complex_complex(G,A4) ) )
=> ( groups3708469109370488835omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8778_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T3: set_complex,S3: set_complex,I: fun_complex_complex,J: fun_complex_complex,T4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,S5)
=> ( aa_set_complex_o(finite3207457112153483333omplex,T3)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(S3,S5))
=> ( aa_complex_complex(I,aa_complex_complex(J,A4)) = A4 ) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(S3,S5))
=> aa_set_complex_o(member_complex(aa_complex_complex(J,A4)),minus_811609699411566653omplex(T4,T3)) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T4,T3))
=> ( aa_complex_complex(J,aa_complex_complex(I,B4)) = B4 ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T4,T3))
=> aa_set_complex_o(member_complex(aa_complex_complex(I,B4)),minus_811609699411566653omplex(S3,S5)) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),S5)
=> ( aa_complex_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),T3)
=> ( aa_complex_complex(H,B4) = one_one_complex ) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),S3)
=> ( aa_complex_complex(H,aa_complex_complex(J,A4)) = aa_complex_complex(G,A4) ) )
=> ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8779_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_real,H: fun_real_real] :
( aa_set_real_o(finite_finite_real,S5)
=> ( aa_set_real_o(finite_finite_real,T3)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S5))
=> aa_set_real_o(member_real(aa_real_real(J,A4)),minus_minus_set_real(T4,T3)) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(T4,T3))
=> aa_set_real_o(member_real(aa_real_real(I,B4)),minus_minus_set_real(S3,S5)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S5)
=> ( aa_real_real(G,A4) = one_one_real ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),T3)
=> ( aa_real_real(H,B4) = one_one_real ) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),S3)
=> ( aa_real_real(H,aa_real_real(J,A4)) = aa_real_real(G,A4) ) )
=> ( groups1681761925125756287l_real(G,S3) = groups1681761925125756287l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_8780_le__floor__iff,axiom,
! [Z: int,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).
% le_floor_iff
tff(fact_8781_le__floor__iff,axiom,
! [Z: int,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).
% le_floor_iff
tff(fact_8782_floor__less__iff,axiom,
! [Xb: real,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).
% floor_less_iff
tff(fact_8783_floor__less__iff,axiom,
! [Xb: rat,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Z)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).
% floor_less_iff
tff(fact_8784_le__floor__add,axiom,
! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))) ).
% le_floor_add
tff(fact_8785_le__floor__add,axiom,
! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(Xb),Y))) ).
% le_floor_add
tff(fact_8786_int__add__floor,axiom,
! [Z: int,Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,Xb)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),Xb)) ).
% int_add_floor
tff(fact_8787_int__add__floor,axiom,
! [Z: int,Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,Xb)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),Xb)) ).
% int_add_floor
tff(fact_8788_floor__add__int,axiom,
! [Xb: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) ).
% floor_add_int
tff(fact_8789_floor__add__int,axiom,
! [Xb: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Z))) ).
% floor_add_int
tff(fact_8790_floor__power,axiom,
! [Xb: real,Nb: nat] :
( ( Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)) )
=> ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,Xb)),Nb) ) ) ).
% floor_power
tff(fact_8791_floor__power,axiom,
! [Xb: rat,Nb: nat] :
( ( Xb = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb)) )
=> ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(Xb),Nb)) = aa_nat_int(power_power_int(aa_rat_int(archim3151403230148437115or_rat,Xb)),Nb) ) ) ).
% floor_power
tff(fact_8792_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_8793_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_8794_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(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_8795_gbinomial__of__nat__symmetric,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).
% gbinomial_of_nat_symmetric
tff(fact_8796_prod_Osetdiff__irrelevant,axiom,
! [A3: set_int,G: fun_int_complex] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups7440179247065528705omplex(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ok(G)))) = groups7440179247065528705omplex(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8797_prod_Osetdiff__irrelevant,axiom,
! [A3: set_complex,G: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ol(G)))) = groups3708469109370488835omplex(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8798_prod_Osetdiff__irrelevant,axiom,
! [A3: set_int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups2316167850115554303t_real(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_om(G)))) = groups2316167850115554303t_real(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8799_prod_Osetdiff__irrelevant,axiom,
! [A3: set_complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_on(G)))) = groups766887009212190081x_real(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8800_prod_Osetdiff__irrelevant,axiom,
! [A3: set_int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_oo(G)))) = groups1072433553688619179nt_rat(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8801_prod_Osetdiff__irrelevant,axiom,
! [A3: set_complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_op(G)))) = groups225925009352817453ex_rat(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8802_prod_Osetdiff__irrelevant,axiom,
! [A3: set_int,G: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_oq(G)))) = groups1707563613775114915nt_nat(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8803_prod_Osetdiff__irrelevant,axiom,
! [A3: set_complex,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_or(G)))) = groups861055069439313189ex_nat(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8804_prod_Osetdiff__irrelevant,axiom,
! [A3: set_complex,G: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups858564598930262913ex_int(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_os(G)))) = groups858564598930262913ex_int(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8805_prod_Osetdiff__irrelevant,axiom,
! [A3: set_nat,G: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups6464643781859351333omplex(G,minus_minus_set_nat(A3,collect_nat(aTP_Lamm_ot(G)))) = groups6464643781859351333omplex(G,A3) ) ) ).
% prod.setdiff_irrelevant
tff(fact_8806_prod_Onat__diff__reindex,axiom,
! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_je(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% prod.nat_diff_reindex
tff(fact_8807_prod_Onat__diff__reindex,axiom,
! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ou(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% prod.nat_diff_reindex
tff(fact_8808_prod_OatLeastAtMost__rev,axiom,
! [G: fun_nat_nat,Nb: nat,Mb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_go(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).
% prod.atLeastAtMost_rev
tff(fact_8809_prod_OatLeastAtMost__rev,axiom,
! [G: fun_nat_int,Nb: nat,Mb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ov(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).
% prod.atLeastAtMost_rev
tff(fact_8810_gbinomial__Suc,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(aTP_Lamm_ow(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K))) ).
% gbinomial_Suc
tff(fact_8811_gbinomial__Suc,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_ox(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).
% gbinomial_Suc
tff(fact_8812_gbinomial__Suc,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(divide_divide_real,groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_oy,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).
% gbinomial_Suc
tff(fact_8813_gbinomial__Suc,axiom,
! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oz,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).
% gbinomial_Suc
tff(fact_8814_gbinomial__Suc,axiom,
! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(divide_divide_int(groups705719431365010083at_int(aTP_Lamm_pa(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).
% gbinomial_Suc
tff(fact_8815_less__1__prod2,axiom,
! [I5: set_real,I: real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,I5)
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I))
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8816_less__1__prod2,axiom,
! [I5: set_nat,I: nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( aa_set_nat_o(member_nat(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I))
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8817_less__1__prod2,axiom,
! [I5: set_int,I: int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,I5)
=> ( aa_set_int_o(member_int(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8818_less__1__prod2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I))
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8819_less__1__prod2,axiom,
! [I5: set_real,I: real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,I5)
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I))
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8820_less__1__prod2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I))
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8821_less__1__prod2,axiom,
! [I5: set_real,I: real,F: fun_real_real] :
( aa_set_real_o(finite_finite_real,I5)
=> ( aa_set_real_o(member_real(I),I5)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I))
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8822_less__1__prod2,axiom,
! [I5: set_nat,I: nat,F: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( aa_set_nat_o(member_nat(I),I5)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I))
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups129246275422532515t_real(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8823_less__1__prod2,axiom,
! [I5: set_int,I: int,F: fun_int_real] :
( aa_set_int_o(finite_finite_int,I5)
=> ( aa_set_int_o(member_int(I),I5)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I))
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8824_less__1__prod2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( aa_set_complex_o(member_complex(I),I5)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I))
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ) ).
% less_1_prod2
tff(fact_8825_less__1__prod,axiom,
! [I5: set_complex,F: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8826_less__1__prod,axiom,
! [I5: set_nat,F: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups129246275422532515t_real(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8827_less__1__prod,axiom,
! [I5: set_int,F: fun_int_real] :
( aa_set_int_o(finite_finite_int,I5)
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8828_less__1__prod,axiom,
! [I5: set_real,F: fun_real_real] :
( aa_set_real_o(finite_finite_real,I5)
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I3)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8829_less__1__prod,axiom,
! [I5: set_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8830_less__1__prod,axiom,
! [I5: set_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,I5)
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8831_less__1__prod,axiom,
! [I5: set_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,I5)
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8832_less__1__prod,axiom,
! [I5: set_real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,I5)
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I3)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8833_less__1__prod,axiom,
! [I5: set_complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,I5)
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8834_less__1__prod,axiom,
! [I5: set_real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,I5)
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),I5)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I3)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I5)) ) ) ) ).
% less_1_prod
tff(fact_8835_prod_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_real] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups2316167850115554303t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A3,B3))),groups2316167850115554303t_real(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8836_prod_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_real] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups766887009212190081x_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A3,B3))),groups766887009212190081x_real(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8837_prod_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_rat] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups1072433553688619179nt_rat(G,A3) = aa_rat_rat(times_times_rat(groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,B3))),groups1072433553688619179nt_rat(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8838_prod_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_rat] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups225925009352817453ex_rat(G,A3) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A3,B3))),groups225925009352817453ex_rat(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8839_prod_Osubset__diff,axiom,
! [B3: set_int,A3: set_int,G: fun_int_nat] :
( aa_set_int_o(ord_less_eq_set_int(B3),A3)
=> ( aa_set_int_o(finite_finite_int,A3)
=> ( groups1707563613775114915nt_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A3,B3))),groups1707563613775114915nt_nat(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8840_prod_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_nat] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups861055069439313189ex_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A3,B3))),groups861055069439313189ex_nat(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8841_prod_Osubset__diff,axiom,
! [B3: set_complex,A3: set_complex,G: fun_complex_int] :
( aa_set_complex_o(ord_le211207098394363844omplex(B3),A3)
=> ( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups858564598930262913ex_int(G,A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A3,B3))),groups858564598930262913ex_int(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8842_prod_Osubset__diff,axiom,
! [B3: set_nat,A3: set_nat,G: fun_nat_real] :
( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups129246275422532515t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,minus_minus_set_nat(A3,B3))),groups129246275422532515t_real(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8843_prod_Osubset__diff,axiom,
! [B3: set_nat,A3: set_nat,G: fun_nat_rat] :
( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups73079841787564623at_rat(G,A3) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A3,B3))),groups73079841787564623at_rat(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8844_prod_Osubset__diff,axiom,
! [B3: set_nat,A3: set_nat,G: fun_nat_nat] :
( aa_set_nat_o(ord_less_eq_set_nat(B3),A3)
=> ( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups708209901874060359at_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,minus_minus_set_nat(A3,B3))),groups708209901874060359at_nat(G,B3)) ) ) ) ).
% prod.subset_diff
tff(fact_8845_prod_Omono__neutral__cong__right,axiom,
! [T4: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_complex(G,X4) = one_one_complex ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
=> ( groups713298508707869441omplex(G,T4) = groups713298508707869441omplex(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8846_prod_Omono__neutral__cong__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_complex,H: fun_int_complex] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_complex(G,X4) = one_one_complex ) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
=> ( groups7440179247065528705omplex(G,T4) = groups7440179247065528705omplex(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8847_prod_Omono__neutral__cong__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_complex(G,X4) = one_one_complex ) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
=> ( groups3708469109370488835omplex(G,T4) = groups3708469109370488835omplex(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8848_prod_Omono__neutral__cong__right,axiom,
! [T4: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_real(G,X4) = one_one_real ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
=> ( groups1681761925125756287l_real(G,T4) = groups1681761925125756287l_real(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8849_prod_Omono__neutral__cong__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_real(G,X4) = one_one_real ) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
=> ( groups2316167850115554303t_real(G,T4) = groups2316167850115554303t_real(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8850_prod_Omono__neutral__cong__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_real(G,X4) = one_one_real ) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
=> ( groups766887009212190081x_real(G,T4) = groups766887009212190081x_real(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8851_prod_Omono__neutral__cong__right,axiom,
! [T4: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_rat(G,X4) = one_one_rat ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
=> ( groups4061424788464935467al_rat(G,T4) = groups4061424788464935467al_rat(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8852_prod_Omono__neutral__cong__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_rat(G,X4) = one_one_rat ) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
=> ( groups1072433553688619179nt_rat(G,T4) = groups1072433553688619179nt_rat(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8853_prod_Omono__neutral__cong__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_rat(G,X4) = one_one_rat ) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
=> ( groups225925009352817453ex_rat(G,T4) = groups225925009352817453ex_rat(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8854_prod_Omono__neutral__cong__right,axiom,
! [T4: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_nat(G,X4) = one_one_nat ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
=> ( groups4696554848551431203al_nat(G,T4) = groups4696554848551431203al_nat(H,S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_8855_prod_Omono__neutral__cong__left,axiom,
! [T4: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_complex(H,X4) = one_one_complex ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
=> ( groups713298508707869441omplex(G,S3) = groups713298508707869441omplex(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8856_prod_Omono__neutral__cong__left,axiom,
! [T4: set_int,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_complex(H,X4) = one_one_complex ) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
=> ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8857_prod_Omono__neutral__cong__left,axiom,
! [T4: set_complex,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_complex(H,X4) = one_one_complex ) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
=> ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8858_prod_Omono__neutral__cong__left,axiom,
! [T4: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_real(H,X4) = one_one_real ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
=> ( groups1681761925125756287l_real(G,S3) = groups1681761925125756287l_real(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8859_prod_Omono__neutral__cong__left,axiom,
! [T4: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_real(H,X4) = one_one_real ) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
=> ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8860_prod_Omono__neutral__cong__left,axiom,
! [T4: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_real(H,X4) = one_one_real ) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
=> ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8861_prod_Omono__neutral__cong__left,axiom,
! [T4: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_rat(H,X4) = one_one_rat ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
=> ( groups4061424788464935467al_rat(G,S3) = groups4061424788464935467al_rat(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8862_prod_Omono__neutral__cong__left,axiom,
! [T4: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_rat(H,X4) = one_one_rat ) )
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),S3)
=> ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
=> ( groups1072433553688619179nt_rat(G,S3) = groups1072433553688619179nt_rat(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8863_prod_Omono__neutral__cong__left,axiom,
! [T4: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_rat(H,X4) = one_one_rat ) )
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),S3)
=> ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
=> ( groups225925009352817453ex_rat(G,S3) = groups225925009352817453ex_rat(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8864_prod_Omono__neutral__cong__left,axiom,
! [T4: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
( aa_set_real_o(finite_finite_real,T4)
=> ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),minus_minus_set_real(T4,S3))
=> ( aa_real_nat(H,X4) = one_one_nat ) )
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),S3)
=> ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
=> ( groups4696554848551431203al_nat(G,S3) = groups4696554848551431203al_nat(H,T4) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_8865_prod_Omono__neutral__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_complex] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_complex(G,X4) = one_one_complex ) )
=> ( groups7440179247065528705omplex(G,T4) = groups7440179247065528705omplex(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8866_prod_Omono__neutral__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_complex(G,X4) = one_one_complex ) )
=> ( groups3708469109370488835omplex(G,T4) = groups3708469109370488835omplex(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8867_prod_Omono__neutral__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_real(G,X4) = one_one_real ) )
=> ( groups2316167850115554303t_real(G,T4) = groups2316167850115554303t_real(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8868_prod_Omono__neutral__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_real(G,X4) = one_one_real ) )
=> ( groups766887009212190081x_real(G,T4) = groups766887009212190081x_real(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8869_prod_Omono__neutral__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_rat(G,X4) = one_one_rat ) )
=> ( groups1072433553688619179nt_rat(G,T4) = groups1072433553688619179nt_rat(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8870_prod_Omono__neutral__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_rat(G,X4) = one_one_rat ) )
=> ( groups225925009352817453ex_rat(G,T4) = groups225925009352817453ex_rat(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8871_prod_Omono__neutral__right,axiom,
! [T4: set_int,S3: set_int,G: fun_int_nat] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_nat(G,X4) = one_one_nat ) )
=> ( groups1707563613775114915nt_nat(G,T4) = groups1707563613775114915nt_nat(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8872_prod_Omono__neutral__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_nat(G,X4) = one_one_nat ) )
=> ( groups861055069439313189ex_nat(G,T4) = groups861055069439313189ex_nat(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8873_prod_Omono__neutral__right,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_int(G,X4) = one_one_int ) )
=> ( groups858564598930262913ex_int(G,T4) = groups858564598930262913ex_int(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8874_prod_Omono__neutral__right,axiom,
! [T4: set_nat,S3: set_nat,G: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,T4)
=> ( aa_set_nat_o(ord_less_eq_set_nat(S3),T4)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T4,S3))
=> ( aa_nat_complex(G,X4) = one_one_complex ) )
=> ( groups6464643781859351333omplex(G,T4) = groups6464643781859351333omplex(G,S3) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_8875_prod_Omono__neutral__left,axiom,
! [T4: set_int,S3: set_int,G: fun_int_complex] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_complex(G,X4) = one_one_complex ) )
=> ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8876_prod_Omono__neutral__left,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_complex(G,X4) = one_one_complex ) )
=> ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8877_prod_Omono__neutral__left,axiom,
! [T4: set_int,S3: set_int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_real(G,X4) = one_one_real ) )
=> ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8878_prod_Omono__neutral__left,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_real(G,X4) = one_one_real ) )
=> ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8879_prod_Omono__neutral__left,axiom,
! [T4: set_int,S3: set_int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_rat(G,X4) = one_one_rat ) )
=> ( groups1072433553688619179nt_rat(G,S3) = groups1072433553688619179nt_rat(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8880_prod_Omono__neutral__left,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_rat(G,X4) = one_one_rat ) )
=> ( groups225925009352817453ex_rat(G,S3) = groups225925009352817453ex_rat(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8881_prod_Omono__neutral__left,axiom,
! [T4: set_int,S3: set_int,G: fun_int_nat] :
( aa_set_int_o(finite_finite_int,T4)
=> ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(T4,S3))
=> ( aa_int_nat(G,X4) = one_one_nat ) )
=> ( groups1707563613775114915nt_nat(G,S3) = groups1707563613775114915nt_nat(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8882_prod_Omono__neutral__left,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_nat(G,X4) = one_one_nat ) )
=> ( groups861055069439313189ex_nat(G,S3) = groups861055069439313189ex_nat(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8883_prod_Omono__neutral__left,axiom,
! [T4: set_complex,S3: set_complex,G: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,T4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
=> ( ! [X4: complex] :
( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T4,S3))
=> ( aa_complex_int(G,X4) = one_one_int ) )
=> ( groups858564598930262913ex_int(G,S3) = groups858564598930262913ex_int(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8884_prod_Omono__neutral__left,axiom,
! [T4: set_nat,S3: set_nat,G: fun_nat_complex] :
( aa_set_nat_o(finite_finite_nat,T4)
=> ( aa_set_nat_o(ord_less_eq_set_nat(S3),T4)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T4,S3))
=> ( aa_nat_complex(G,X4) = one_one_complex ) )
=> ( groups6464643781859351333omplex(G,S3) = groups6464643781859351333omplex(G,T4) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_8885_prod_Osame__carrierI,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_complex,H: fun_real_complex] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_complex(H,B4) = one_one_complex ) )
=> ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
=> ( groups713298508707869441omplex(G,A3) = groups713298508707869441omplex(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8886_prod_Osame__carrierI,axiom,
! [C4: set_int,A3: set_int,B3: set_int,G: fun_int_complex,H: fun_int_complex] :
( aa_set_int_o(finite_finite_int,C4)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
=> ( aa_set_int_o(ord_less_eq_set_int(B3),C4)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(C4,A3))
=> ( aa_int_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(C4,B3))
=> ( aa_int_complex(H,B4) = one_one_complex ) )
=> ( ( groups7440179247065528705omplex(G,C4) = groups7440179247065528705omplex(H,C4) )
=> ( groups7440179247065528705omplex(G,A3) = groups7440179247065528705omplex(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8887_prod_Osame__carrierI,axiom,
! [C4: set_complex,A3: set_complex,B3: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C4)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C4,A3))
=> ( aa_complex_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C4,B3))
=> ( aa_complex_complex(H,B4) = one_one_complex ) )
=> ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
=> ( groups3708469109370488835omplex(G,A3) = groups3708469109370488835omplex(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8888_prod_Osame__carrierI,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_real,H: fun_real_real] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_real(G,A4) = one_one_real ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_real(H,B4) = one_one_real ) )
=> ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
=> ( groups1681761925125756287l_real(G,A3) = groups1681761925125756287l_real(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8889_prod_Osame__carrierI,axiom,
! [C4: set_int,A3: set_int,B3: set_int,G: fun_int_real,H: fun_int_real] :
( aa_set_int_o(finite_finite_int,C4)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
=> ( aa_set_int_o(ord_less_eq_set_int(B3),C4)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(C4,A3))
=> ( aa_int_real(G,A4) = one_one_real ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(C4,B3))
=> ( aa_int_real(H,B4) = one_one_real ) )
=> ( ( groups2316167850115554303t_real(G,C4) = groups2316167850115554303t_real(H,C4) )
=> ( groups2316167850115554303t_real(G,A3) = groups2316167850115554303t_real(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8890_prod_Osame__carrierI,axiom,
! [C4: set_complex,A3: set_complex,B3: set_complex,G: fun_complex_real,H: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C4)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C4,A3))
=> ( aa_complex_real(G,A4) = one_one_real ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C4,B3))
=> ( aa_complex_real(H,B4) = one_one_real ) )
=> ( ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) )
=> ( groups766887009212190081x_real(G,A3) = groups766887009212190081x_real(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8891_prod_Osame__carrierI,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_rat,H: fun_real_rat] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_rat(G,A4) = one_one_rat ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_rat(H,B4) = one_one_rat ) )
=> ( ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) )
=> ( groups4061424788464935467al_rat(G,A3) = groups4061424788464935467al_rat(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8892_prod_Osame__carrierI,axiom,
! [C4: set_int,A3: set_int,B3: set_int,G: fun_int_rat,H: fun_int_rat] :
( aa_set_int_o(finite_finite_int,C4)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
=> ( aa_set_int_o(ord_less_eq_set_int(B3),C4)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(C4,A3))
=> ( aa_int_rat(G,A4) = one_one_rat ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(C4,B3))
=> ( aa_int_rat(H,B4) = one_one_rat ) )
=> ( ( groups1072433553688619179nt_rat(G,C4) = groups1072433553688619179nt_rat(H,C4) )
=> ( groups1072433553688619179nt_rat(G,A3) = groups1072433553688619179nt_rat(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8893_prod_Osame__carrierI,axiom,
! [C4: set_complex,A3: set_complex,B3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C4)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C4,A3))
=> ( aa_complex_rat(G,A4) = one_one_rat ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C4,B3))
=> ( aa_complex_rat(H,B4) = one_one_rat ) )
=> ( ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) )
=> ( groups225925009352817453ex_rat(G,A3) = groups225925009352817453ex_rat(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8894_prod_Osame__carrierI,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_nat,H: fun_real_nat] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_nat(G,A4) = one_one_nat ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_nat(H,B4) = one_one_nat ) )
=> ( ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) )
=> ( groups4696554848551431203al_nat(G,A3) = groups4696554848551431203al_nat(H,B3) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_8895_prod_Osame__carrier,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_complex,H: fun_real_complex] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_complex(H,B4) = one_one_complex ) )
=> ( ( groups713298508707869441omplex(G,A3) = groups713298508707869441omplex(H,B3) )
<=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8896_prod_Osame__carrier,axiom,
! [C4: set_int,A3: set_int,B3: set_int,G: fun_int_complex,H: fun_int_complex] :
( aa_set_int_o(finite_finite_int,C4)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
=> ( aa_set_int_o(ord_less_eq_set_int(B3),C4)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(C4,A3))
=> ( aa_int_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(C4,B3))
=> ( aa_int_complex(H,B4) = one_one_complex ) )
=> ( ( groups7440179247065528705omplex(G,A3) = groups7440179247065528705omplex(H,B3) )
<=> ( groups7440179247065528705omplex(G,C4) = groups7440179247065528705omplex(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8897_prod_Osame__carrier,axiom,
! [C4: set_complex,A3: set_complex,B3: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
( aa_set_complex_o(finite3207457112153483333omplex,C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C4)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C4,A3))
=> ( aa_complex_complex(G,A4) = one_one_complex ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C4,B3))
=> ( aa_complex_complex(H,B4) = one_one_complex ) )
=> ( ( groups3708469109370488835omplex(G,A3) = groups3708469109370488835omplex(H,B3) )
<=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8898_prod_Osame__carrier,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_real,H: fun_real_real] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_real(G,A4) = one_one_real ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_real(H,B4) = one_one_real ) )
=> ( ( groups1681761925125756287l_real(G,A3) = groups1681761925125756287l_real(H,B3) )
<=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8899_prod_Osame__carrier,axiom,
! [C4: set_int,A3: set_int,B3: set_int,G: fun_int_real,H: fun_int_real] :
( aa_set_int_o(finite_finite_int,C4)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
=> ( aa_set_int_o(ord_less_eq_set_int(B3),C4)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(C4,A3))
=> ( aa_int_real(G,A4) = one_one_real ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(C4,B3))
=> ( aa_int_real(H,B4) = one_one_real ) )
=> ( ( groups2316167850115554303t_real(G,A3) = groups2316167850115554303t_real(H,B3) )
<=> ( groups2316167850115554303t_real(G,C4) = groups2316167850115554303t_real(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8900_prod_Osame__carrier,axiom,
! [C4: set_complex,A3: set_complex,B3: set_complex,G: fun_complex_real,H: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C4)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C4,A3))
=> ( aa_complex_real(G,A4) = one_one_real ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C4,B3))
=> ( aa_complex_real(H,B4) = one_one_real ) )
=> ( ( groups766887009212190081x_real(G,A3) = groups766887009212190081x_real(H,B3) )
<=> ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8901_prod_Osame__carrier,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_rat,H: fun_real_rat] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_rat(G,A4) = one_one_rat ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_rat(H,B4) = one_one_rat ) )
=> ( ( groups4061424788464935467al_rat(G,A3) = groups4061424788464935467al_rat(H,B3) )
<=> ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8902_prod_Osame__carrier,axiom,
! [C4: set_int,A3: set_int,B3: set_int,G: fun_int_rat,H: fun_int_rat] :
( aa_set_int_o(finite_finite_int,C4)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
=> ( aa_set_int_o(ord_less_eq_set_int(B3),C4)
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),minus_minus_set_int(C4,A3))
=> ( aa_int_rat(G,A4) = one_one_rat ) )
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(C4,B3))
=> ( aa_int_rat(H,B4) = one_one_rat ) )
=> ( ( groups1072433553688619179nt_rat(G,A3) = groups1072433553688619179nt_rat(H,B3) )
<=> ( groups1072433553688619179nt_rat(G,C4) = groups1072433553688619179nt_rat(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8903_prod_Osame__carrier,axiom,
! [C4: set_complex,A3: set_complex,B3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C4)
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C4,A3))
=> ( aa_complex_rat(G,A4) = one_one_rat ) )
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C4,B3))
=> ( aa_complex_rat(H,B4) = one_one_rat ) )
=> ( ( groups225925009352817453ex_rat(G,A3) = groups225925009352817453ex_rat(H,B3) )
<=> ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8904_prod_Osame__carrier,axiom,
! [C4: set_real,A3: set_real,B3: set_real,G: fun_real_nat,H: fun_real_nat] :
( aa_set_real_o(finite_finite_real,C4)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
=> ( aa_set_real_o(ord_less_eq_set_real(B3),C4)
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),minus_minus_set_real(C4,A3))
=> ( aa_real_nat(G,A4) = one_one_nat ) )
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(C4,B3))
=> ( aa_real_nat(H,B4) = one_one_nat ) )
=> ( ( groups4696554848551431203al_nat(G,A3) = groups4696554848551431203al_nat(H,B3) )
<=> ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_8905_one__add__floor,axiom,
! [Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).
% one_add_floor
tff(fact_8906_one__add__floor,axiom,
! [Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) ).
% one_add_floor
tff(fact_8907_prod_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).
% prod.atLeast0_atMost_Suc
tff(fact_8908_prod_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).
% prod.atLeast0_atMost_Suc
tff(fact_8909_prod_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).
% prod.atLeast0_atMost_Suc
tff(fact_8910_prod_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).
% prod.atLeast0_atMost_Suc
tff(fact_8911_prod_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_8912_prod_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_8913_prod_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_8914_prod_Onat__ivl__Suc_H,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_8915_prod_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_8916_prod_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Mb)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_8917_prod_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_8918_prod_OatLeast__Suc__atMost,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_8919_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_8920_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_8921_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).
% gbinomial_addition_formula
tff(fact_8922_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_8923_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_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(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).
% gbinomial_absorb_comp
tff(fact_8924_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_8925_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(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_8926_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_8927_gbinomial__mult__1,axiom,
! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(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_8928_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_8929_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_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_8930_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(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_8931_floor__eq,axiom,
! [Nb: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
=> ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).
% floor_eq
tff(fact_8932_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).
% real_of_int_floor_add_one_gt
tff(fact_8933_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).
% real_of_int_floor_gt_diff_one
tff(fact_8934_prod_OlessThan__Suc__shift,axiom,
! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_gh(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% prod.lessThan_Suc_shift
tff(fact_8935_prod_OlessThan__Suc__shift,axiom,
! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_gq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% prod.lessThan_Suc_shift
tff(fact_8936_prod_OlessThan__Suc__shift,axiom,
! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_gg(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% prod.lessThan_Suc_shift
tff(fact_8937_prod_OlessThan__Suc__shift,axiom,
! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_gr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% prod.lessThan_Suc_shift
tff(fact_8938_prod_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),groups129246275422532515t_real(aTP_Lamm_gh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_8939_prod_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Mb)),groups73079841787564623at_rat(aTP_Lamm_gq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_8940_prod_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),groups708209901874060359at_nat(aTP_Lamm_gg(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_8941_prod_OSuc__reindex__ivl,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(aTP_Lamm_gr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_8942_prod_OatLeast1__atMost__eq,axiom,
! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups708209901874060359at_nat(aTP_Lamm_gg(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% prod.atLeast1_atMost_eq
tff(fact_8943_prod_OatLeast1__atMost__eq,axiom,
! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups705719431365010083at_int(aTP_Lamm_gr(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).
% prod.atLeast1_atMost_eq
tff(fact_8944_fact__prod,axiom,
! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,groups708209901874060359at_nat(aTP_Lamm_aj,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).
% fact_prod
tff(fact_8945_fact__prod,axiom,
! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,groups708209901874060359at_nat(aTP_Lamm_aj,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).
% fact_prod
tff(fact_8946_fact__prod,axiom,
! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,groups708209901874060359at_nat(aTP_Lamm_aj,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).
% fact_prod
tff(fact_8947_prod__mono__strict,axiom,
! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_complex )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8948_prod__mono__strict,axiom,
! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ! [I3: nat] :
( aa_set_nat_o(member_nat(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_nat )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8949_prod__mono__strict,axiom,
! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_int )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8950_prod__mono__strict,axiom,
! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),A3)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_real )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8951_prod__mono__strict,axiom,
! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_complex )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups861055069439313189ex_nat(F,A3)),groups861055069439313189ex_nat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8952_prod__mono__strict,axiom,
! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ! [I3: int] :
( aa_set_int_o(member_int(I3),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_int )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8953_prod__mono__strict,axiom,
! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) ) )
=> ( ( A3 != bot_bot_set_real )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4696554848551431203al_nat(F,A3)),groups4696554848551431203al_nat(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8954_prod__mono__strict,axiom,
! [A3: set_complex,F: fun_complex_int,G: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I3))
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) ) )
=> ( ( A3 != bot_bot_set_complex )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8955_prod__mono__strict,axiom,
! [A3: set_real,F: fun_real_int,G: fun_real_int] :
( aa_set_real_o(finite_finite_real,A3)
=> ( ! [I3: real] :
( aa_set_real_o(member_real(I3),A3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3))
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,I3)),aa_real_int(G,I3)) ) )
=> ( ( A3 != bot_bot_set_real )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8956_prod__mono__strict,axiom,
! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ! [I3: complex] :
( aa_set_complex_o(member_complex(I3),A3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,I3)),aa_complex_real(G,I3)) ) )
=> ( ( A3 != bot_bot_set_complex )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(G,A3)) ) ) ) ).
% prod_mono_strict
tff(fact_8957_even__prod__iff,axiom,
! [A3: set_nat,F: fun_nat_Code_integer] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3455450783089532116nteger(F,A3))
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8958_even__prod__iff,axiom,
! [A3: set_int,F: fun_int_Code_integer] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3827104343326376752nteger(F,A3))
<=> ? [X: int] :
( aa_set_int_o(member_int(X),A3)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8959_even__prod__iff,axiom,
! [A3: set_complex,F: fun_co8153928105989009288nteger] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A3))
<=> ? [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8960_even__prod__iff,axiom,
! [A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups1707563613775114915nt_nat(F,A3))
<=> ? [X: int] :
( aa_set_int_o(member_int(X),A3)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8961_even__prod__iff,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A3))
<=> ? [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8962_even__prod__iff,axiom,
! [A3: set_complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A3))
<=> ? [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8963_even__prod__iff,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups708209901874060359at_nat(F,A3))
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8964_even__prod__iff,axiom,
! [A3: set_nat,F: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A3))
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8965_even__prod__iff,axiom,
! [A3: set_int,F: fun_int_int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups1705073143266064639nt_int(F,A3))
<=> ? [X: int] :
( aa_set_int_o(member_int(X),A3)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(F,X)) ) ) ) ).
% even_prod_iff
tff(fact_8966_floor__unique,axiom,
! [Z: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb) = Z ) ) ) ).
% floor_unique
tff(fact_8967_floor__unique,axiom,
! [Z: int,Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
=> ( aa_rat_int(archim3151403230148437115or_rat,Xb) = Z ) ) ) ).
% floor_unique
tff(fact_8968_floor__eq__iff,axiom,
! [Xb: real,A: int] :
( ( aa_real_int(archim6058952711729229775r_real,Xb) = A )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_8969_floor__eq__iff,axiom,
! [Xb: rat,A: int] :
( ( aa_rat_int(archim3151403230148437115or_rat,Xb) = A )
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).
% floor_eq_iff
tff(fact_8970_floor__split,axiom,
! [P: fun_int_o,Ta: real] :
( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Ta))
<=> ! [I4: int] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),Ta)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)) )
=> aa_int_o(P,I4) ) ) ).
% floor_split
tff(fact_8971_floor__split,axiom,
! [P: fun_int_o,Ta: rat] :
( aa_int_o(P,aa_rat_int(archim3151403230148437115or_rat,Ta))
<=> ! [I4: int] :
( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),Ta)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)) )
=> aa_int_o(P,I4) ) ) ).
% floor_split
tff(fact_8972_prod_Oremove,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups2703838992350267259T_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xb)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% prod.remove
tff(fact_8973_prod_Oremove,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> ( groups766887009212190081x_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xb)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).
% prod.remove
tff(fact_8974_prod_Oremove,axiom,
! [A3: set_int,Xb: int,G: fun_int_real] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(member_int(Xb),A3)
=> ( groups2316167850115554303t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xb)),groups2316167850115554303t_real(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ) ).
% prod.remove
tff(fact_8975_prod_Oremove,axiom,
! [A3: set_real,Xb: real,G: fun_real_real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( aa_set_real_o(member_real(Xb),A3)
=> ( groups1681761925125756287l_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xb)),groups1681761925125756287l_real(G,minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)))) ) ) ) ).
% prod.remove
tff(fact_8976_prod_Oremove,axiom,
! [A3: set_nat,Xb: nat,G: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_set_nat_o(member_nat(Xb),A3)
=> ( groups129246275422532515t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xb)),groups129246275422532515t_real(G,minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat)))) ) ) ) ).
% prod.remove
tff(fact_8977_prod_Oremove,axiom,
! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3)
=> ( groups5726676334696518183BT_rat(G,A3) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,Xb)),groups5726676334696518183BT_rat(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% prod.remove
tff(fact_8978_prod_Oremove,axiom,
! [A3: set_complex,Xb: complex,G: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_set_complex_o(member_complex(Xb),A3)
=> ( groups225925009352817453ex_rat(G,A3) = aa_rat_rat(times_times_rat(aa_complex_rat(G,Xb)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).
% prod.remove
tff(fact_8979_prod_Oremove,axiom,
! [A3: set_int,Xb: int,G: fun_int_rat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_set_int_o(member_int(Xb),A3)
=> ( groups1072433553688619179nt_rat(G,A3) = aa_rat_rat(times_times_rat(aa_int_rat(G,Xb)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ) ).
% prod.remove
tff(fact_8980_prod_Oremove,axiom,
! [A3: set_real,Xb: real,G: fun_real_rat] :
( aa_set_real_o(finite_finite_real,A3)
=> ( aa_set_real_o(member_real(Xb),A3)
=> ( groups4061424788464935467al_rat(G,A3) = aa_rat_rat(times_times_rat(aa_real_rat(G,Xb)),groups4061424788464935467al_rat(G,minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)))) ) ) ) ).
% prod.remove
tff(fact_8981_prod_Oremove,axiom,
! [A3: set_nat,Xb: nat,G: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_set_nat_o(member_nat(Xb),A3)
=> ( groups73079841787564623at_rat(G,A3) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Xb)),groups73079841787564623at_rat(G,minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat)))) ) ) ) ).
% prod.remove
tff(fact_8982_prod_Oinsert__remove,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xb)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% prod.insert_remove
tff(fact_8983_prod_Oinsert__remove,axiom,
! [A3: set_complex,G: fun_complex_real,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups766887009212190081x_real(G,insert_complex(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xb)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).
% prod.insert_remove
tff(fact_8984_prod_Oinsert__remove,axiom,
! [A3: set_int,G: fun_int_real,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups2316167850115554303t_real(G,insert_int(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xb)),groups2316167850115554303t_real(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ).
% prod.insert_remove
tff(fact_8985_prod_Oinsert__remove,axiom,
! [A3: set_real,G: fun_real_real,Xb: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups1681761925125756287l_real(G,insert_real(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xb)),groups1681761925125756287l_real(G,minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)))) ) ) ).
% prod.insert_remove
tff(fact_8986_prod_Oinsert__remove,axiom,
! [A3: set_nat,G: fun_nat_real,Xb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups129246275422532515t_real(G,insert_nat(Xb,A3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xb)),groups129246275422532515t_real(G,minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat)))) ) ) ).
% prod.insert_remove
tff(fact_8987_prod_Oinsert__remove,axiom,
! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xb: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
=> ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(Xb,A3)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,Xb)),groups5726676334696518183BT_rat(G,minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% prod.insert_remove
tff(fact_8988_prod_Oinsert__remove,axiom,
! [A3: set_complex,G: fun_complex_rat,Xb: complex] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( groups225925009352817453ex_rat(G,insert_complex(Xb,A3)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,Xb)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).
% prod.insert_remove
tff(fact_8989_prod_Oinsert__remove,axiom,
! [A3: set_int,G: fun_int_rat,Xb: int] :
( aa_set_int_o(finite_finite_int,A3)
=> ( groups1072433553688619179nt_rat(G,insert_int(Xb,A3)) = aa_rat_rat(times_times_rat(aa_int_rat(G,Xb)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)))) ) ) ).
% prod.insert_remove
tff(fact_8990_prod_Oinsert__remove,axiom,
! [A3: set_real,G: fun_real_rat,Xb: real] :
( aa_set_real_o(finite_finite_real,A3)
=> ( groups4061424788464935467al_rat(G,insert_real(Xb,A3)) = aa_rat_rat(times_times_rat(aa_real_rat(G,Xb)),groups4061424788464935467al_rat(G,minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)))) ) ) ).
% prod.insert_remove
tff(fact_8991_prod_Oinsert__remove,axiom,
! [A3: set_nat,G: fun_nat_rat,Xb: nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( groups73079841787564623at_rat(G,insert_nat(Xb,A3)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Xb)),groups73079841787564623at_rat(G,minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat)))) ) ) ).
% prod.insert_remove
tff(fact_8992_le__mult__floor,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
=> aa_int_o(aa_int_fun_int_o(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_8993_le__mult__floor,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))) ) ) ).
% le_mult_floor
tff(fact_8994_less__floor__iff,axiom,
! [Z: int,Xb: real] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(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)),Xb) ) ).
% less_floor_iff
tff(fact_8995_less__floor__iff,axiom,
! [Z: int,Xb: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,Xb))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb) ) ).
% less_floor_iff
tff(fact_8996_floor__le__iff,axiom,
! [Xb: real,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_8997_floor__le__iff,axiom,
! [Xb: rat,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Z)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).
% floor_le_iff
tff(fact_8998_floor__correct,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int))) ) ).
% floor_correct
tff(fact_8999_floor__correct,axiom,
! [Xb: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb))),Xb)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int))) ) ).
% floor_correct
tff(fact_9000_prod_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% prod.ub_add_nat
tff(fact_9001_prod_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% prod.ub_add_nat
tff(fact_9002_prod_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% prod.ub_add_nat
tff(fact_9003_prod_Oub__add__nat,axiom,
! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).
% prod.ub_add_nat
tff(fact_9004_fold__atLeastAtMost__nat_Osimps,axiom,
! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ).
% fold_atLeastAtMost_nat.simps
tff(fact_9005_fold__atLeastAtMost__nat_Oelims,axiom,
! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
=> ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).
% fold_atLeastAtMost_nat.elims
tff(fact_9006_floor__eq2,axiom,
! [Nb: int,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
=> ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).
% floor_eq2
tff(fact_9007_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_9008_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(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).
% Suc_times_gbinomial
tff(fact_9009_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_9010_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_9011_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(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).
% gbinomial_absorption
tff(fact_9012_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_9013_prod_Odelta__remove,axiom,
! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_hh(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(B,A)),groups2703838992350267259T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups2703838992350267259T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% prod.delta_remove
tff(fact_9014_prod_Odelta__remove,axiom,
! [S3: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_hi(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B,A)),groups766887009212190081x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups766887009212190081x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).
% prod.delta_remove
tff(fact_9015_prod_Odelta__remove,axiom,
! [S3: set_int,A: int,B: fun_int_real,C: fun_int_real] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_hp(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(B,A)),groups2316167850115554303t_real(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups2316167850115554303t_real(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).
% prod.delta_remove
tff(fact_9016_prod_Odelta__remove,axiom,
! [S3: set_real,A: real,B: fun_real_real,C: fun_real_real] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_pb(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B,A)),groups1681761925125756287l_real(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups1681761925125756287l_real(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).
% prod.delta_remove
tff(fact_9017_prod_Odelta__remove,axiom,
! [S3: set_nat,A: nat,B: fun_nat_real,C: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_pc(A),B),C),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(B,A)),groups129246275422532515t_real(C,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat)))),groups129246275422532515t_real(C,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat)))) ) ) ).
% prod.delta_remove
tff(fact_9018_prod_Odelta__remove,axiom,
! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
=> ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_hj(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(B,A)),groups5726676334696518183BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups5726676334696518183BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% prod.delta_remove
tff(fact_9019_prod_Odelta__remove,axiom,
! [S3: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,S3)
=> ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_hk(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_rat_rat(times_times_rat(aa_complex_rat(B,A)),groups225925009352817453ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups225925009352817453ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).
% prod.delta_remove
tff(fact_9020_prod_Odelta__remove,axiom,
! [S3: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
( aa_set_int_o(finite_finite_int,S3)
=> ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_hq(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_rat_rat(times_times_rat(aa_int_rat(B,A)),groups1072433553688619179nt_rat(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups1072433553688619179nt_rat(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).
% prod.delta_remove
tff(fact_9021_prod_Odelta__remove,axiom,
! [S3: set_real,A: real,B: fun_real_rat,C: fun_real_rat] :
( aa_set_real_o(finite_finite_real,S3)
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_pd(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_rat_rat(times_times_rat(aa_real_rat(B,A)),groups4061424788464935467al_rat(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups4061424788464935467al_rat(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).
% prod.delta_remove
tff(fact_9022_prod_Odelta__remove,axiom,
! [S3: set_nat,A: nat,B: fun_nat_rat,C: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_pe(A),B),C),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_rat_rat(times_times_rat(aa_nat_rat(B,A)),groups73079841787564623at_rat(C,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat)))),groups73079841787564623at_rat(C,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat)))) ) ) ).
% prod.delta_remove
tff(fact_9023_gbinomial__trinomial__revision,axiom,
! [K: nat,Mb: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_9024_gbinomial__trinomial__revision,axiom,
! [K: nat,Mb: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),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,Mb),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_9025_fact__eq__fact__times,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),groups708209901874060359at_nat(aTP_Lamm_aj,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),Mb))) ) ) ).
% fact_eq_fact_times
tff(fact_9026_prod__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_rat] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B4)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9027_prod__mono2,axiom,
! [B3: set_int,A3: set_int,F: fun_int_rat] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B4)) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9028_prod__mono2,axiom,
! [B3: set_complex,A3: set_complex,F: fun_complex_rat] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B4)) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9029_prod__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_int] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,B4)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,A4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9030_prod__mono2,axiom,
! [B3: set_complex,A3: set_complex,F: fun_complex_int] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,B4)) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),A3)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A4)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9031_prod__mono2,axiom,
! [B3: set_real,A3: set_real,F: fun_real_real] :
( aa_set_real_o(finite_finite_real,B3)
=> ( aa_set_real_o(ord_less_eq_set_real(A3),B3)
=> ( ! [B4: real] :
( aa_set_real_o(member_real(B4),minus_minus_set_real(B3,A3))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B4)) )
=> ( ! [A4: real] :
( aa_set_real_o(member_real(A4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9032_prod__mono2,axiom,
! [B3: set_int,A3: set_int,F: fun_int_real] :
( aa_set_int_o(finite_finite_int,B3)
=> ( aa_set_int_o(ord_less_eq_set_int(A3),B3)
=> ( ! [B4: int] :
( aa_set_int_o(member_int(B4),minus_minus_set_int(B3,A3))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B4)) )
=> ( ! [A4: int] :
( aa_set_int_o(member_int(A4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9033_prod__mono2,axiom,
! [B3: set_complex,A3: set_complex,F: fun_complex_real] :
( aa_set_complex_o(finite3207457112153483333omplex,B3)
=> ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B3)
=> ( ! [B4: complex] :
( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B3,A3))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,B4)) )
=> ( ! [A4: complex] :
( aa_set_complex_o(member_complex(A4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9034_prod__mono2,axiom,
! [B3: set_nat,A3: set_nat,F: fun_nat_rat] :
( aa_set_nat_o(finite_finite_nat,B3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
=> ( ! [B4: nat] :
( aa_set_nat_o(member_nat(B4),minus_minus_set_nat(B3,A3))
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B4)) )
=> ( ! [A4: nat] :
( aa_set_nat_o(member_nat(A4),A3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A4)) )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9035_prod__mono2,axiom,
! [B3: set_nat,A3: set_nat,F: fun_nat_real] :
( aa_set_nat_o(finite_finite_nat,B3)
=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B3)
=> ( ! [B4: nat] :
( aa_set_nat_o(member_nat(B4),minus_minus_set_nat(B3,A3))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,B4)) )
=> ( ! [A4: nat] :
( aa_set_nat_o(member_nat(A4),A3)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A4)) )
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A3)),groups129246275422532515t_real(F,B3)) ) ) ) ) ).
% prod_mono2
tff(fact_9036_floor__divide__lower,axiom,
! [Q3: real,P2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
=> aa_real_o(aa_real_fun_real_o(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),Q3)))),Q3)),P2) ) ).
% floor_divide_lower
tff(fact_9037_floor__divide__lower,axiom,
! [Q3: rat,P2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),Q3)),P2) ) ).
% floor_divide_lower
tff(fact_9038_gbinomial__code,axiom,
! [A: complex,K: nat] :
aa_nat_complex(gbinomial_complex(A),K) = $ite(K = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_pf(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_9039_gbinomial__code,axiom,
! [A: rat,K: nat] :
aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_pg(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_9040_gbinomial__code,axiom,
! [A: real,K: nat] :
aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ph(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_9041_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_9042_gbinomial__factors,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_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_9043_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_9044_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_9045_gbinomial__rec,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_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)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).
% gbinomial_rec
tff(fact_9046_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_9047_gbinomial__index__swap,axiom,
! [K: nat,Nb: 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,Nb))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),Nb)) ).
% gbinomial_index_swap
tff(fact_9048_gbinomial__index__swap,axiom,
! [K: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),Nb)) ).
% gbinomial_index_swap
tff(fact_9049_gbinomial__index__swap,axiom,
! [K: nat,Nb: 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,Nb))),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)),Nb)),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)),Nb)) ).
% gbinomial_index_swap
tff(fact_9050_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_9051_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(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).
% gbinomial_negated_upper
tff(fact_9052_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_9053_pochhammer__Suc__prod,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_pi(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod
tff(fact_9054_pochhammer__Suc__prod,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_pj,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod
tff(fact_9055_pochhammer__Suc__prod,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pk,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod
tff(fact_9056_pochhammer__Suc__prod,axiom,
! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_pl(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod
tff(fact_9057_pochhammer__prod__rev,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_pm(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).
% pochhammer_prod_rev
tff(fact_9058_pochhammer__prod__rev,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_pn(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).
% pochhammer_prod_rev
tff(fact_9059_pochhammer__prod__rev,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_po(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).
% pochhammer_prod_rev
tff(fact_9060_pochhammer__prod__rev,axiom,
! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pp(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).
% pochhammer_prod_rev
tff(fact_9061_floor__divide__upper,axiom,
! [Q3: real,P2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
=> aa_real_o(aa_real_fun_real_o(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),Q3)))),one_one_real)),Q3)) ) ).
% floor_divide_upper
tff(fact_9062_floor__divide__upper,axiom,
! [Q3: rat,P2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q3)))),one_one_rat)),Q3)) ) ).
% floor_divide_upper
tff(fact_9063_fact__div__fact,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = groups708209901874060359at_nat(aTP_Lamm_aj,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)) ) ) ).
% fact_div_fact
tff(fact_9064_round__def,axiom,
! [Xb: real] : aa_real_int(archim8280529875227126926d_real,Xb) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% round_def
tff(fact_9065_round__def,axiom,
! [Xb: rat] : aa_rat_int(archim7778729529865785530nd_rat,Xb) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).
% round_def
tff(fact_9066_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_9067_gbinomial__minus,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).
% gbinomial_minus
tff(fact_9068_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_9069_prod_Oin__pairs,axiom,
! [G: fun_nat_real,Mb: nat,Nb: nat] : groups129246275422532515t_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))),Mb)),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))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_pq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.in_pairs
tff(fact_9070_prod_Oin__pairs,axiom,
! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups73079841787564623at_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))),Mb)),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))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_pr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.in_pairs
tff(fact_9071_prod_Oin__pairs,axiom,
! [G: fun_nat_nat,Mb: nat,Nb: nat] : groups708209901874060359at_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))),Mb)),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))),Nb)))) = groups708209901874060359at_nat(aTP_Lamm_ps(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.in_pairs
tff(fact_9072_prod_Oin__pairs,axiom,
! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_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))),Mb)),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))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_pt(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).
% prod.in_pairs
tff(fact_9073_gbinomial__reduce__nat,axiom,
! [K: nat,A: complex] :
( aa_nat_o(aa_nat_fun_nat_o(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_9074_gbinomial__reduce__nat,axiom,
! [K: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(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_9075_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_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(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).
% gbinomial_reduce_nat
tff(fact_9076_sum__atLeastAtMost__code,axiom,
! [F: fun_nat_complex,A: nat,B: nat] : groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_pu(F),A,B),zero_zero_complex) ).
% sum_atLeastAtMost_code
tff(fact_9077_sum__atLeastAtMost__code,axiom,
! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_pv(F),A,B),zero_zero_rat) ).
% sum_atLeastAtMost_code
tff(fact_9078_sum__atLeastAtMost__code,axiom,
! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_pw(F),A,B),zero_zero_int) ).
% sum_atLeastAtMost_code
tff(fact_9079_sum__atLeastAtMost__code,axiom,
! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_px(F),A),B),zero_zero_nat) ).
% sum_atLeastAtMost_code
tff(fact_9080_sum__atLeastAtMost__code,axiom,
! [F: fun_nat_real,A: nat,B: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_py(F),A),B),zero_zero_real) ).
% sum_atLeastAtMost_code
tff(fact_9081_pochhammer__Suc__prod__rev,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_pm(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod_rev
tff(fact_9082_pochhammer__Suc__prod__rev,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_pn(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod_rev
tff(fact_9083_pochhammer__Suc__prod__rev,axiom,
! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_po(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod_rev
tff(fact_9084_pochhammer__Suc__prod__rev,axiom,
! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pp(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).
% pochhammer_Suc_prod_rev
tff(fact_9085_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_9086_gbinomial__pochhammer,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).
% gbinomial_pochhammer
tff(fact_9087_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_9088_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_9089_gbinomial__pochhammer_H,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_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_9090_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_9091_gbinomial__sum__up__index,axiom,
! [K: nat,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_pz(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).
% gbinomial_sum_up_index
tff(fact_9092_gbinomial__sum__up__index,axiom,
! [K: nat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_qa(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).
% gbinomial_sum_up_index
tff(fact_9093_gbinomial__sum__up__index,axiom,
! [K: nat,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qb,K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).
% gbinomial_sum_up_index
tff(fact_9094_floor__log__eq__powr__iff,axiom,
! [Xb: real,B: real,K: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = K )
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),Xb)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_9095_gbinomial__absorption_H,axiom,
! [K: nat,A: complex] :
( aa_nat_o(aa_nat_fun_nat_o(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_9096_gbinomial__absorption_H,axiom,
! [K: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_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_9097_gbinomial__absorption_H,axiom,
! [K: nat,A: real] :
( aa_nat_o(aa_nat_fun_nat_o(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_9098_floor__log2__div2,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).
% floor_log2_div2
tff(fact_9099_fact__code,axiom,
! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).
% fact_code
tff(fact_9100_fact__code,axiom,
! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).
% fact_code
tff(fact_9101_fact__code,axiom,
! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).
% fact_code
tff(fact_9102_floor__log__nat__eq__if,axiom,
! [B: nat,Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(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,Nb),one_one_nat)))
=> ( aa_nat_o(aa_nat_fun_nat_o(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,Nb) ) ) ) ) ).
% floor_log_nat_eq_if
tff(fact_9103_gbinomial__partial__row__sum,axiom,
! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_mf(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),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,Mb),one_one_nat))) ).
% gbinomial_partial_row_sum
tff(fact_9104_gbinomial__partial__row__sum,axiom,
! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_mg(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),one_one_rat)),aa_num_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,Mb),one_one_nat))) ).
% gbinomial_partial_row_sum
tff(fact_9105_gbinomial__partial__row__sum,axiom,
! [A: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mh,A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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,Mb)),one_one_real)),aa_num_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,Mb),one_one_nat))) ).
% gbinomial_partial_row_sum
tff(fact_9106_round__altdef,axiom,
! [Xb: real] :
aa_real_int(archim8280529875227126926d_real,Xb) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb),aa_real_int(archim6058952711729229775r_real,Xb)) ).
% round_altdef
tff(fact_9107_round__altdef,axiom,
! [Xb: rat] :
aa_rat_int(archim7778729529865785530nd_rat,Xb) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Xb),aa_rat_int(archim3151403230148437115or_rat,Xb)) ).
% round_altdef
tff(fact_9108_gbinomial__r__part__sum,axiom,
! [Mb: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Mb))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(aa_num_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))),Mb)) ).
% gbinomial_r_part_sum
tff(fact_9109_gbinomial__r__part__sum,axiom,
! [Mb: nat] : groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Mb))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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))),Mb)) ).
% gbinomial_r_part_sum
tff(fact_9110_gbinomial__r__part__sum,axiom,
! [Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),one_one_real)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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))),Mb)) ).
% gbinomial_r_part_sum
tff(fact_9111_Maclaurin__sin__bound,axiom,
! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ku,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).
% Maclaurin_sin_bound
tff(fact_9112_central__binomial__lower__bound,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),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))),Nb)),Nb))) ) ).
% central_binomial_lower_bound
tff(fact_9113_inverse__mult__distrib,axiom,
! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).
% inverse_mult_distrib
tff(fact_9114_inverse__mult__distrib,axiom,
! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).
% inverse_mult_distrib
tff(fact_9115_inverse__mult__distrib,axiom,
! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).
% inverse_mult_distrib
tff(fact_9116_inverse__1,axiom,
aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).
% inverse_1
tff(fact_9117_inverse__1,axiom,
aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).
% inverse_1
tff(fact_9118_inverse__1,axiom,
aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).
% inverse_1
tff(fact_9119_inverse__eq__1__iff,axiom,
! [Xb: real] :
( ( aa_real_real(inverse_inverse_real,Xb) = one_one_real )
<=> ( Xb = one_one_real ) ) ).
% inverse_eq_1_iff
tff(fact_9120_inverse__eq__1__iff,axiom,
! [Xb: complex] :
( ( aa_complex_complex(invers8013647133539491842omplex,Xb) = one_one_complex )
<=> ( Xb = one_one_complex ) ) ).
% inverse_eq_1_iff
tff(fact_9121_inverse__eq__1__iff,axiom,
! [Xb: rat] :
( ( aa_rat_rat(inverse_inverse_rat,Xb) = one_one_rat )
<=> ( Xb = one_one_rat ) ) ).
% inverse_eq_1_iff
tff(fact_9122_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( member_set_nat(I,set_or4236626031148496127et_nat(K))
<=> aa_set_nat_o(ord_less_eq_set_nat(I),K) ) ).
% atMost_iff
tff(fact_9123_atMost__iff,axiom,
! [I: rat,K: rat] :
( aa_set_rat_o(member_rat(I),set_ord_atMost_rat(K))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K) ) ).
% atMost_iff
tff(fact_9124_atMost__iff,axiom,
! [I: num,K: num] :
( aa_set_num_o(member_num(I),set_ord_atMost_num(K))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),K) ) ).
% atMost_iff
tff(fact_9125_atMost__iff,axiom,
! [I: int,K: int] :
( aa_set_int_o(member_int(I),set_ord_atMost_int(K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K) ) ).
% atMost_iff
tff(fact_9126_atMost__iff,axiom,
! [I: real,K: real] :
( aa_set_real_o(member_real(I),set_ord_atMost_real(K))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K) ) ).
% atMost_iff
tff(fact_9127_atMost__iff,axiom,
! [I: nat,K: nat] :
( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ).
% atMost_iff
tff(fact_9128_binomial__Suc__n,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).
% binomial_Suc_n
tff(fact_9129_binomial__n__n,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).
% binomial_n_n
tff(fact_9130_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).
% inverse_nonpositive_iff_nonpositive
tff(fact_9131_inverse__nonpositive__iff__nonpositive,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).
% inverse_nonpositive_iff_nonpositive
tff(fact_9132_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).
% inverse_nonnegative_iff_nonnegative
tff(fact_9133_inverse__nonnegative__iff__nonnegative,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).
% inverse_nonnegative_iff_nonnegative
tff(fact_9134_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% inverse_less_iff_less
tff(fact_9135_inverse__less__iff__less,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% inverse_less_iff_less
tff(fact_9136_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).
% inverse_less_iff_less_neg
tff(fact_9137_inverse__less__iff__less__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).
% inverse_less_iff_less_neg
tff(fact_9138_inverse__negative__iff__negative,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).
% inverse_negative_iff_negative
tff(fact_9139_inverse__negative__iff__negative,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).
% inverse_negative_iff_negative
tff(fact_9140_inverse__positive__iff__positive,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).
% inverse_positive_iff_positive
tff(fact_9141_inverse__positive__iff__positive,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).
% inverse_positive_iff_positive
tff(fact_9142_atMost__subset__iff,axiom,
! [Xb: set_nat,Y: set_nat] :
( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(Xb),set_or4236626031148496127et_nat(Y))
<=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).
% atMost_subset_iff
tff(fact_9143_atMost__subset__iff,axiom,
! [Xb: rat,Y: rat] :
( aa_set_rat_o(ord_less_eq_set_rat(set_ord_atMost_rat(Xb)),set_ord_atMost_rat(Y))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).
% atMost_subset_iff
tff(fact_9144_atMost__subset__iff,axiom,
! [Xb: num,Y: num] :
( aa_set_num_o(ord_less_eq_set_num(set_ord_atMost_num(Xb)),set_ord_atMost_num(Y))
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).
% atMost_subset_iff
tff(fact_9145_atMost__subset__iff,axiom,
! [Xb: int,Y: int] :
( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(Xb)),set_ord_atMost_int(Y))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).
% atMost_subset_iff
tff(fact_9146_atMost__subset__iff,axiom,
! [Xb: real,Y: real] :
( aa_set_real_o(ord_less_eq_set_real(set_ord_atMost_real(Xb)),set_ord_atMost_real(Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).
% atMost_subset_iff
tff(fact_9147_atMost__subset__iff,axiom,
! [Xb: nat,Y: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).
% atMost_subset_iff
tff(fact_9148_binomial__1,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).
% binomial_1
tff(fact_9149_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_9150_binomial__eq__0__iff,axiom,
! [Nb: nat,K: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).
% binomial_eq_0_iff
tff(fact_9151_binomial__Suc__Suc,axiom,
! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).
% binomial_Suc_Suc
tff(fact_9152_binomial__n__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).
% binomial_n_0
tff(fact_9153_prod__eq__1__iff,axiom,
! [A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( ( groups1707563613775114915nt_nat(F,A3) = one_one_nat )
<=> ! [X: int] :
( aa_set_int_o(member_int(X),A3)
=> ( aa_int_nat(F,X) = one_one_nat ) ) ) ) ).
% prod_eq_1_iff
tff(fact_9154_prod__eq__1__iff,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( ( groups861055069439313189ex_nat(F,A3) = one_one_nat )
<=> ! [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
=> ( aa_complex_nat(F,X) = one_one_nat ) ) ) ) ).
% prod_eq_1_iff
tff(fact_9155_prod__eq__1__iff,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( ( groups708209901874060359at_nat(F,A3) = one_one_nat )
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
=> ( aa_nat_nat(F,X) = one_one_nat ) ) ) ) ).
% prod_eq_1_iff
tff(fact_9156_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).
% inverse_le_iff_le_neg
tff(fact_9157_inverse__le__iff__le__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).
% inverse_le_iff_le_neg
tff(fact_9158_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).
% inverse_le_iff_le
tff(fact_9159_inverse__le__iff__le,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).
% inverse_le_iff_le
tff(fact_9160_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).
% left_inverse
tff(fact_9161_left__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).
% left_inverse
tff(fact_9162_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).
% left_inverse
tff(fact_9163_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).
% right_inverse
tff(fact_9164_right__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).
% right_inverse
tff(fact_9165_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).
% right_inverse
tff(fact_9166_inverse__eq__divide__numeral,axiom,
! [W: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,W)) ).
% inverse_eq_divide_numeral
tff(fact_9167_inverse__eq__divide__numeral,axiom,
! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).
% inverse_eq_divide_numeral
tff(fact_9168_inverse__eq__divide__numeral,axiom,
! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,W)) ).
% inverse_eq_divide_numeral
tff(fact_9169_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))
<=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
| aa_set_nat_o(ord_less_eq_set_nat(H),H2) ) ) ).
% Icc_subset_Iic_iff
tff(fact_9170_Icc__subset__Iic__iff,axiom,
! [L: rat,H: rat,H2: rat] :
( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(H2))
<=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
| aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,H),H2) ) ) ).
% Icc_subset_Iic_iff
tff(fact_9171_Icc__subset__Iic__iff,axiom,
! [L: num,H: num,H2: num] :
( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(H2))
<=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
| aa_num_o(aa_num_fun_num_o(ord_less_eq_num,H),H2) ) ) ).
% Icc_subset_Iic_iff
tff(fact_9172_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2))
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,H),H2) ) ) ).
% Icc_subset_Iic_iff
tff(fact_9173_Icc__subset__Iic__iff,axiom,
! [L: int,H: int,H2: int] :
( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H2))
<=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,H),H2) ) ) ).
% Icc_subset_Iic_iff
tff(fact_9174_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H2: real] :
( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H2))
<=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
| aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),H2) ) ) ).
% Icc_subset_Iic_iff
tff(fact_9175_sum_OatMost__Suc,axiom,
! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).
% sum.atMost_Suc
tff(fact_9176_sum_OatMost__Suc,axiom,
! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).
% sum.atMost_Suc
tff(fact_9177_sum_OatMost__Suc,axiom,
! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = 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,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).
% sum.atMost_Suc
tff(fact_9178_sum_OatMost__Suc,axiom,
! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).
% sum.atMost_Suc
tff(fact_9179_zero__less__binomial__iff,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).
% zero_less_binomial_iff
tff(fact_9180_prod_OatMost__Suc,axiom,
! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).
% prod.atMost_Suc
tff(fact_9181_prod_OatMost__Suc,axiom,
! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).
% prod.atMost_Suc
tff(fact_9182_prod_OatMost__Suc,axiom,
! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).
% prod.atMost_Suc
tff(fact_9183_prod_OatMost__Suc,axiom,
! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).
% prod.atMost_Suc
tff(fact_9184_prod__pos__nat__iff,axiom,
! [A3: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A3))
<=> ! [X: int] :
( aa_set_int_o(member_int(X),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X)) ) ) ) ).
% prod_pos_nat_iff
tff(fact_9185_prod__pos__nat__iff,axiom,
! [A3: set_complex,F: fun_complex_nat] :
( aa_set_complex_o(finite3207457112153483333omplex,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A3))
<=> ! [X: complex] :
( aa_set_complex_o(member_complex(X),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X)) ) ) ) ).
% prod_pos_nat_iff
tff(fact_9186_prod__pos__nat__iff,axiom,
! [A3: set_nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,A3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A3))
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),A3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X)) ) ) ) ).
% prod_pos_nat_iff
tff(fact_9187_inverse__eq__divide__neg__numeral,axiom,
! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).
% inverse_eq_divide_neg_numeral
tff(fact_9188_inverse__eq__divide__neg__numeral,axiom,
! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).
% inverse_eq_divide_neg_numeral
tff(fact_9189_inverse__eq__divide__neg__numeral,axiom,
! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).
% inverse_eq_divide_neg_numeral
tff(fact_9190_power__inverse,axiom,
! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,A)),Nb) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).
% power_inverse
tff(fact_9191_power__inverse,axiom,
! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),Nb) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),Nb)) ).
% power_inverse
tff(fact_9192_power__inverse,axiom,
! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),Nb) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),Nb)) ).
% power_inverse
tff(fact_9193_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: real,Xb: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_9194_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: complex,Xb: complex] :
( ( aa_complex_complex(times_times_complex(Y),Xb) = aa_complex_complex(times_times_complex(Xb),Y) )
=> ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_9195_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: rat,Xb: rat] :
( ( aa_rat_rat(times_times_rat(Y),Xb) = aa_rat_rat(times_times_rat(Xb),Y) )
=> ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Y)),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_9196_sum__choose__upper,axiom,
! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qc,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).
% sum_choose_upper
tff(fact_9197_choose__one,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).
% choose_one
tff(fact_9198_atMost__def,axiom,
! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_qd(U)) ).
% atMost_def
tff(fact_9199_atMost__def,axiom,
! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_qe,U)) ).
% atMost_def
tff(fact_9200_atMost__def,axiom,
! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_qf,U)) ).
% atMost_def
tff(fact_9201_atMost__def,axiom,
! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_qg,U)) ).
% atMost_def
tff(fact_9202_atMost__def,axiom,
! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_qh,U)) ).
% atMost_def
tff(fact_9203_atMost__def,axiom,
! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bi,U)) ).
% atMost_def
tff(fact_9204_sum__choose__lower,axiom,
! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).
% sum_choose_lower
tff(fact_9205_choose__rising__sum_I2_J,axiom,
! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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,Nb),Mb)),one_one_nat)),Mb) ).
% choose_rising_sum(2)
tff(fact_9206_choose__rising__sum_I1_J,axiom,
! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).
% choose_rising_sum(1)
tff(fact_9207_norm__inverse__le__norm,axiom,
! [R2: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,Xb))),aa_real_real(inverse_inverse_real,R2)) ) ) ).
% norm_inverse_le_norm
tff(fact_9208_norm__inverse__le__norm,axiom,
! [R2: real,Xb: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,Xb))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,Xb))),aa_real_real(inverse_inverse_real,R2)) ) ) ).
% norm_inverse_le_norm
tff(fact_9209_binomial__eq__0,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).
% binomial_eq_0
tff(fact_9210_positive__imp__inverse__positive,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).
% positive_imp_inverse_positive
tff(fact_9211_positive__imp__inverse__positive,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).
% positive_imp_inverse_positive
tff(fact_9212_negative__imp__inverse__negative,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).
% negative_imp_inverse_negative
tff(fact_9213_negative__imp__inverse__negative,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).
% negative_imp_inverse_negative
tff(fact_9214_inverse__positive__imp__positive,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
=> ( ( A != zero_zero_real )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).
% inverse_positive_imp_positive
tff(fact_9215_inverse__positive__imp__positive,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
=> ( ( A != zero_zero_rat )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).
% inverse_positive_imp_positive
tff(fact_9216_inverse__negative__imp__negative,axiom,
! [A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
=> ( ( A != zero_zero_real )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).
% inverse_negative_imp_negative
tff(fact_9217_inverse__negative__imp__negative,axiom,
! [A: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
=> ( ( A != zero_zero_rat )
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).
% inverse_negative_imp_negative
tff(fact_9218_less__imp__inverse__less__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).
% less_imp_inverse_less_neg
tff(fact_9219_less__imp__inverse__less__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).
% less_imp_inverse_less_neg
tff(fact_9220_inverse__less__imp__less__neg,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).
% inverse_less_imp_less_neg
tff(fact_9221_inverse__less__imp__less__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).
% inverse_less_imp_less_neg
tff(fact_9222_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).
% less_imp_inverse_less
tff(fact_9223_less__imp__inverse__less,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).
% less_imp_inverse_less
tff(fact_9224_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).
% inverse_less_imp_less
tff(fact_9225_inverse__less__imp__less,axiom,
! [A: rat,B: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).
% inverse_less_imp_less
tff(fact_9226_nonzero__inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_9227_nonzero__inverse__mult__distrib,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_9228_nonzero__inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_9229_Suc__times__binomial,axiom,
! [K: nat,Nb: 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,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).
% Suc_times_binomial
tff(fact_9230_Suc__times__binomial__eq,axiom,
! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).
% Suc_times_binomial_eq
tff(fact_9231_inverse__numeral__1,axiom,
aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).
% inverse_numeral_1
tff(fact_9232_inverse__numeral__1,axiom,
aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).
% inverse_numeral_1
tff(fact_9233_inverse__numeral__1,axiom,
aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_num_rat(numeral_numeral_rat,one) ).
% inverse_numeral_1
tff(fact_9234_inverse__unique,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = one_one_real )
=> ( aa_real_real(inverse_inverse_real,A) = B ) ) ).
% inverse_unique
tff(fact_9235_inverse__unique,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
=> ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).
% inverse_unique
tff(fact_9236_inverse__unique,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(A),B) = one_one_rat )
=> ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).
% inverse_unique
tff(fact_9237_field__class_Ofield__divide__inverse,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).
% field_class.field_divide_inverse
tff(fact_9238_field__class_Ofield__divide__inverse,axiom,
! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).
% field_class.field_divide_inverse
tff(fact_9239_field__class_Ofield__divide__inverse,axiom,
! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).
% field_class.field_divide_inverse
tff(fact_9240_divide__inverse,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).
% divide_inverse
tff(fact_9241_divide__inverse,axiom,
! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).
% divide_inverse
tff(fact_9242_divide__inverse,axiom,
! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).
% divide_inverse
tff(fact_9243_divide__inverse__commute,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),A) ).
% divide_inverse_commute
tff(fact_9244_divide__inverse__commute,axiom,
! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).
% divide_inverse_commute
tff(fact_9245_divide__inverse__commute,axiom,
! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),A) ).
% divide_inverse_commute
tff(fact_9246_inverse__eq__divide,axiom,
! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ).
% inverse_eq_divide
tff(fact_9247_inverse__eq__divide,axiom,
! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).
% inverse_eq_divide
tff(fact_9248_inverse__eq__divide,axiom,
! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ).
% inverse_eq_divide
tff(fact_9249_binomial__symmetric,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).
% binomial_symmetric
tff(fact_9250_power__mult__power__inverse__commute,axiom,
! [Xb: real,Mb: nat,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).
% power_mult_power_inverse_commute
tff(fact_9251_power__mult__power__inverse__commute,axiom,
! [Xb: complex,Mb: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).
% power_mult_power_inverse_commute
tff(fact_9252_power__mult__power__inverse__commute,axiom,
! [Xb: rat,Mb: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).
% power_mult_power_inverse_commute
tff(fact_9253_power__mult__inverse__distrib,axiom,
! [Xb: real,Mb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).
% power_mult_inverse_distrib
tff(fact_9254_power__mult__inverse__distrib,axiom,
! [Xb: complex,Mb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).
% power_mult_inverse_distrib
tff(fact_9255_power__mult__inverse__distrib,axiom,
! [Xb: rat,Mb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(inverse_inverse_rat,Xb)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Xb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).
% power_mult_inverse_distrib
tff(fact_9256_choose__mult__lemma,axiom,
! [Mb: nat,R2: 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,Mb),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),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,Mb),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).
% choose_mult_lemma
tff(fact_9257_mult__inverse__of__nat__commute,axiom,
! [Xaa: nat,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))) ).
% mult_inverse_of_nat_commute
tff(fact_9258_mult__inverse__of__nat__commute,axiom,
! [Xaa: nat,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))) ).
% mult_inverse_of_nat_commute
tff(fact_9259_mult__inverse__of__nat__commute,axiom,
! [Xaa: nat,Xb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))) ).
% mult_inverse_of_nat_commute
tff(fact_9260_binomial__le__pow,axiom,
! [R2: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).
% binomial_le_pow
tff(fact_9261_mult__inverse__of__int__commute,axiom,
! [Xaa: int,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))) ).
% mult_inverse_of_int_commute
tff(fact_9262_mult__inverse__of__int__commute,axiom,
! [Xaa: int,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))) ).
% mult_inverse_of_int_commute
tff(fact_9263_mult__inverse__of__int__commute,axiom,
! [Xaa: int,Xb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))) ).
% mult_inverse_of_int_commute
tff(fact_9264_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_9265_sum__choose__diagonal,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).
% sum_choose_diagonal
tff(fact_9266_vandermonde,axiom,
! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ql(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).
% vandermonde
tff(fact_9267_divide__real__def,axiom,
! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ).
% divide_real_def
tff(fact_9268_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_9269_prod_Otriangle__reindex__eq,axiom,
! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_qm(Nb)))) = groups708209901874060359at_nat(aTP_Lamm_qo(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).
% prod.triangle_reindex_eq
tff(fact_9270_prod_Otriangle__reindex__eq,axiom,
! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_qm(Nb)))) = groups705719431365010083at_int(aTP_Lamm_qq(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).
% prod.triangle_reindex_eq
tff(fact_9271_choose__row__sum,axiom,
! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).
% choose_row_sum
tff(fact_9272_binomial,axiom,
! [A: nat,B: nat,Nb: 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)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qr(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).
% binomial
tff(fact_9273_frac__ge__0,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xb)) ).
% frac_ge_0
tff(fact_9274_frac__ge__0,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xb)) ).
% frac_ge_0
tff(fact_9275_frac__lt__1,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xb)),one_one_real) ).
% frac_lt_1
tff(fact_9276_frac__lt__1,axiom,
! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,Xb)),one_one_rat) ).
% frac_lt_1
tff(fact_9277_frac__1__eq,axiom,
! [Xb: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,Xb) ).
% frac_1_eq
tff(fact_9278_zero__less__binomial,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).
% zero_less_binomial
tff(fact_9279_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_9280_binomial__Suc__Suc__eq__times,axiom,
! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),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,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).
% binomial_Suc_Suc_eq_times
tff(fact_9281_choose__mult,axiom,
! [K: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ) ).
% choose_mult
tff(fact_9282_binomial__absorb__comp,axiom,
! [Nb: 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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).
% binomial_absorb_comp
tff(fact_9283_choose__square__sum,axiom,
! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qs,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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))),Nb)),Nb) ).
% choose_square_sum
tff(fact_9284_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_9285_binomial__absorption,axiom,
! [K: nat,Nb: 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,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).
% binomial_absorption
tff(fact_9286_binomial__r__part__sum,axiom,
! [Mb: 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))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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))),Mb)) ).
% binomial_r_part_sum
tff(fact_9287_choose__linear__sum,axiom,
! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qt,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Nb),one_one_nat))) ).
% choose_linear_sum
tff(fact_9288_forall__pos__mono__1,axiom,
! [P: fun_real_o,E: real] :
( ! [D4: real,E2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E2)
=> ( aa_real_o(P,D4)
=> aa_real_o(P,E2) ) )
=> ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
=> aa_real_o(P,E) ) ) ) ).
% forall_pos_mono_1
tff(fact_9289_forall__pos__mono,axiom,
! [P: fun_real_o,E: real] :
( ! [D4: real,E2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E2)
=> ( aa_real_o(P,D4)
=> aa_real_o(P,E2) ) )
=> ( ! [N: nat] :
( ( N != zero_zero_nat )
=> aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
=> aa_real_o(P,E) ) ) ) ).
% forall_pos_mono
tff(fact_9290_real__arch__inverse,axiom,
! [E: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
<=> ? [N3: nat] :
( ( N3 != zero_zero_nat )
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3)))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E) ) ) ).
% real_arch_inverse
tff(fact_9291_binomial__fact__lemma,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( 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,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).
% binomial_fact_lemma
tff(fact_9292_ln__inverse,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xb)) ) ) ).
% ln_inverse
tff(fact_9293_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_ak,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_9294_binomial__maximum_H,axiom,
! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(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))),Nb)),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))),Nb)),Nb)) ).
% binomial_maximum'
tff(fact_9295_binomial__mono,axiom,
! [K: nat,K6: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
=> ( aa_nat_o(aa_nat_fun_nat_o(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)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).
% binomial_mono
tff(fact_9296_binomial__antimono,axiom,
! [K: nat,K6: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).
% binomial_antimono
tff(fact_9297_binomial__maximum,axiom,
! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% binomial_maximum
tff(fact_9298_binomial__le__pow2,axiom,
! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% binomial_le_pow2
tff(fact_9299_choose__reduce__nat,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),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,Nb),one_one_nat)),K)) ) ) ) ).
% choose_reduce_nat
tff(fact_9300_times__binomial__minus1__eq,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% times_binomial_minus1_eq
tff(fact_9301_binomial__altdef__nat,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),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,Nb),K)))) ) ) ).
% binomial_altdef_nat
tff(fact_9302_log__inverse,axiom,
! [A: real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( ( A != one_one_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) ) ) ) ) ).
% log_inverse
tff(fact_9303_atLeast1__atMost__eq__remove0,axiom,
! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).
% atLeast1_atMost_eq_remove0
tff(fact_9304_binomial__less__binomial__Suc,axiom,
! [K: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).
% binomial_less_binomial_Suc
tff(fact_9305_binomial__strict__antimono,axiom,
! [K: nat,K6: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).
% binomial_strict_antimono
tff(fact_9306_binomial__strict__mono,axiom,
! [K: nat,K6: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
=> ( aa_nat_o(aa_nat_fun_nat_o(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)),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).
% binomial_strict_mono
tff(fact_9307_central__binomial__odd,axiom,
! [Nb: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% central_binomial_odd
tff(fact_9308_binomial__addition__formula,axiom,
! [Nb: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),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,Nb),one_one_nat)),K)) ) ) ).
% binomial_addition_formula
tff(fact_9309_exp__plus__inverse__exp,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).
% exp_plus_inverse_exp
tff(fact_9310_choose__two,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% choose_two
tff(fact_9311_plus__inverse__ge__2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).
% plus_inverse_ge_2
tff(fact_9312_real__inv__sqrt__pow2,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).
% real_inv_sqrt_pow2
tff(fact_9313_tan__cot,axiom,
! [Xb: 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).
% tan_cot
tff(fact_9314_polynomial__product__nat,axiom,
! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
( ! [I3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
=> ( aa_nat_nat(A,I3) = zero_zero_nat ) )
=> ( ! [J2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_qu(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qu(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qw(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).
% polynomial_product_nat
tff(fact_9315_real__le__x__sinh,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).
% real_le_x_sinh
tff(fact_9316_real__le__abs__sinh,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),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,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% real_le_abs_sinh
tff(fact_9317_binomial__code,axiom,
! [Nb: nat,K: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
zero_zero_nat,
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).
% binomial_code
tff(fact_9318_of__nat__id,axiom,
! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).
% of_nat_id
tff(fact_9319_real__scaleR__def,axiom,
! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xb) ).
% real_scaleR_def
tff(fact_9320_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
=> ~ ! [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
=> ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T2)),aa_real_real(sin_real,T2)) ) ) ) ) ).
% complex_unimodular_polar
tff(fact_9321_cot__less__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).
% cot_less_zero
tff(fact_9322_sinh__real__less__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).
% sinh_real_less_iff
tff(fact_9323_sinh__real__neg__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).
% sinh_real_neg_iff
tff(fact_9324_sinh__real__pos__iff,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xb))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).
% sinh_real_pos_iff
tff(fact_9325_cot__npi,axiom,
! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).
% cot_npi
tff(fact_9326_cot__periodic,axiom,
! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).
% cot_periodic
tff(fact_9327_complex__scaleR,axiom,
! [R2: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).
% complex_scaleR
tff(fact_9328_complex__of__real__mult__Complex,axiom,
! [R2: real,Xb: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).
% complex_of_real_mult_Complex
tff(fact_9329_Complex__mult__complex__of__real,axiom,
! [Xb: real,Y: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).
% Complex_mult_complex_of_real
tff(fact_9330_sinh__less__cosh__real,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).
% sinh_less_cosh_real
tff(fact_9331_cosh__real__pos,axiom,
! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).
% cosh_real_pos
tff(fact_9332_cosh__real__strict__mono,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y)) ) ) ).
% cosh_real_strict_mono
tff(fact_9333_cosh__real__nonneg__less__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).
% cosh_real_nonneg_less_iff
tff(fact_9334_cosh__real__nonpos__less__iff,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).
% cosh_real_nonpos_less_iff
tff(fact_9335_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_9336_complex__norm,axiom,
! [Xb: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),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,Xb),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_9337_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_9338_cosh__ln__real,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).
% cosh_ln_real
tff(fact_9339_cot__gt__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).
% cot_gt_zero
tff(fact_9340_sinh__ln__real,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).
% sinh_ln_real
tff(fact_9341_tan__cot_H,axiom,
! [Xb: 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).
% tan_cot'
tff(fact_9342_arctan__def,axiom,
! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_qx,Y)) ).
% arctan_def
tff(fact_9343_arcsin__def,axiom,
! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_qy,Y)) ).
% arcsin_def
tff(fact_9344_signed__take__bit__eq__take__bit__minus,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).
% signed_take_bit_eq_take_bit_minus
tff(fact_9345_modulo__int__unfold,axiom,
! [K: int,Mb: nat,L: int,Nb: nat] :
aa_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,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( Nb = zero_zero_nat ) ),
aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb)),
$ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,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,Mb),Nb))),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,Nb),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).
% modulo_int_unfold
tff(fact_9346_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K)
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
tff(fact_9347_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K)
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
tff(fact_9348_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
tff(fact_9349_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2)))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
tff(fact_9350_signed__take__bit__negative__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),zero_zero_int)
<=> aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).
% signed_take_bit_negative_iff
tff(fact_9351_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,Nb: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
<=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).
% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9352_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,Nb: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
<=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).
% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9353_bit__minus__numeral__int_I1_J,axiom,
! [W: num,Nb: num] :
( aa_nat_o(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,Nb))
<=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).
% bit_minus_numeral_int(1)
tff(fact_9354_bit__minus__numeral__int_I2_J,axiom,
! [W: num,Nb: num] :
( aa_nat_o(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,Nb))
<=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).
% bit_minus_numeral_int(2)
tff(fact_9355_complex__exp__exists,axiom,
! [Z: complex] :
? [A4: complex,R3: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A4)) ).
% complex_exp_exists
tff(fact_9356_divide__complex__def,axiom,
! [Xb: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ).
% divide_complex_def
tff(fact_9357_int__sgnE,axiom,
! [K: int] :
~ ! [N: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% int_sgnE
tff(fact_9358_zsgn__def,axiom,
! [I: int] :
aa_int_int(sgn_sgn_int,I) = $ite(
I = zero_zero_int,
zero_zero_int,
$ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).
% zsgn_def
tff(fact_9359_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V != zero_zero_int )
=> ( aa_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(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).
% div_sgn_abs_cancel
tff(fact_9360_bit__imp__take__bit__positive,axiom,
! [Nb: nat,Mb: nat,K: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Mb),K)) ) ) ).
% bit_imp_take_bit_positive
tff(fact_9361_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
=> ( aa_int_int(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(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).
% div_dvd_sgn_abs
tff(fact_9362_bit__concat__bit__iff,axiom,
! [Mb: nat,K: int,L: int,Nb: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(Mb,K),L)),Nb)
<=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
& aa_nat_o(bit_se1146084159140164899it_int(K),Nb) )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).
% bit_concat_bit_iff
tff(fact_9363_int__bit__bound,axiom,
! [K: int] :
~ ! [N: nat] :
( ! [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M3)
=> ( aa_nat_o(bit_se1146084159140164899it_int(K),M3)
<=> aa_nat_o(bit_se1146084159140164899it_int(K),N) ) )
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
<=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N) ) ) ) ).
% int_bit_bound
tff(fact_9364_bit__int__def,axiom,
! [K: int,Nb: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).
% bit_int_def
tff(fact_9365_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q3: int] :
( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),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,Q3),L)),R2) )
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_9366_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(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) ) )
=> ( ! [Q2: int] :
( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int) )
=> ( ( A22 != zero_zero_int )
=> ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
=> ~ ! [R3: int,Q2: int] :
( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R3) )
=> ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
=> ( aa_int_o(aa_int_fun_int_o(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,Q2),A22)),R3) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_9367_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(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) )
& aa_int_o(aa_int_fun_int_o(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_9368_set__bit__eq,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),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_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).
% set_bit_eq
tff(fact_9369_unset__bit__eq,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),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_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).
% unset_bit_eq
tff(fact_9370_pi__half,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_qz) ).
% pi_half
tff(fact_9371_pi__def,axiom,
pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_qz)) ).
% pi_def
tff(fact_9372_take__bit__Suc__from__most,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).
% take_bit_Suc_from_most
tff(fact_9373_divide__int__unfold,axiom,
! [K: int,Mb: nat,L: int,Nb: nat] :
aa_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,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( Nb = zero_zero_nat ) ),
zero_zero_int,
$ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),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,Mb),Nb)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).
% divide_int_unfold
tff(fact_9374_modulo__int__def,axiom,
! [K: int,L: int] :
aa_int_int(modulo_modulo_int(K),L) = $ite(
L = zero_zero_int,
K,
$ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,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(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_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(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_9375_divide__int__def,axiom,
! [K: int,L: int] :
aa_int_int(divide_divide_int(K),L) = $ite(
L = zero_zero_int,
zero_zero_int,
$ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_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)))),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_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).
% divide_int_def
tff(fact_9376_powr__int,axiom,
! [Xb: real,I: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).
% powr_int
tff(fact_9377_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_9378_nat__1,axiom,
aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).
% nat_1
tff(fact_9379_zless__nat__conj,axiom,
! [W: int,Z: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
& aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).
% zless_nat_conj
tff(fact_9380_zero__less__nat__eq,axiom,
! [Z: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).
% zero_less_nat_eq
tff(fact_9381_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_9382_numeral__power__eq__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,Y: int] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).
% numeral_power_eq_nat_cancel_iff
tff(fact_9383_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,Xb: num,Nb: nat] :
( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).
% nat_eq_numeral_power_cancel_iff
tff(fact_9384_nat__ceiling__le__eq,axiom,
! [Xb: real,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).
% nat_ceiling_le_eq
tff(fact_9385_one__less__nat__eq,axiom,
! [Z: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% one_less_nat_eq
tff(fact_9386_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_9387_numeral__power__less__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_less_nat_cancel_iff
tff(fact_9388_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% nat_less_numeral_power_cancel_iff
tff(fact_9389_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,Xb: num,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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,Xb)),Nb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).
% nat_le_numeral_power_cancel_iff
tff(fact_9390_numeral__power__le__nat__cancel__iff,axiom,
! [Xb: num,Nb: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).
% numeral_power_le_nat_cancel_iff
tff(fact_9391_not__bit__Suc__0__Suc,axiom,
! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).
% not_bit_Suc_0_Suc
tff(fact_9392_bit__Suc__0__iff,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
<=> ( Nb = zero_zero_nat ) ) ).
% bit_Suc_0_iff
tff(fact_9393_nat__zero__as__int,axiom,
zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).
% nat_zero_as_int
tff(fact_9394_nat__numeral__as__int,axiom,
! [X3: num] : aa_num_nat(numeral_numeral_nat,X3) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X3)) ).
% nat_numeral_as_int
tff(fact_9395_nat__mono,axiom,
! [Xb: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).
% nat_mono
tff(fact_9396_nat__one__as__int,axiom,
one_one_nat = aa_int_nat(nat2,one_one_int) ).
% nat_one_as_int
tff(fact_9397_not__bit__Suc__0__numeral,axiom,
! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).
% not_bit_Suc_0_numeral
tff(fact_9398_nat__mono__iff,axiom,
! [Z: int,W: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).
% nat_mono_iff
tff(fact_9399_zless__nat__eq__int__zless,axiom,
! [Mb: nat,Z: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).
% zless_nat_eq_int_zless
tff(fact_9400_nat__le__iff,axiom,
! [Xb: int,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).
% nat_le_iff
tff(fact_9401_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_9402_int__minus,axiom,
! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = 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,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)))) ).
% int_minus
tff(fact_9403_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_9404_nat__plus__as__int,axiom,
! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_plus_as_int
tff(fact_9405_nat__times__as__int,axiom,
! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_times_as_int
tff(fact_9406_nat__minus__as__int,axiom,
! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_minus_as_int
tff(fact_9407_nat__div__as__int,axiom,
! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_div_as_int
tff(fact_9408_nat__mod__as__int,axiom,
! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_mod_as_int
tff(fact_9409_sgn__real__def,axiom,
! [A: real] :
aa_real_real(sgn_sgn_real,A) = $ite(
A = zero_zero_real,
zero_zero_real,
$ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).
% sgn_real_def
tff(fact_9410_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).
% nat_less_eq_zless
tff(fact_9411_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).
% nat_le_eq_zle
tff(fact_9412_split__nat,axiom,
! [P: fun_nat_o,I: int] :
( aa_nat_o(P,aa_int_nat(nat2,I))
<=> ( ! [N3: nat] :
( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
=> aa_nat_o(P,N3) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
=> aa_nat_o(P,zero_zero_nat) ) ) ) ).
% split_nat
tff(fact_9413_le__nat__iff,axiom,
! [K: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).
% le_nat_iff
tff(fact_9414_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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_9415_Suc__as__int,axiom,
! [X3: nat] : aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ).
% Suc_as_int
tff(fact_9416_nat__mult__distrib,axiom,
! [Z: int,Z5: int] :
( aa_int_o(aa_int_fun_int_o(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_9417_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,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_9418_nat__power__eq,axiom,
! [Z: int,Nb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).
% nat_power_eq
tff(fact_9419_floor__eq3,axiom,
! [Nb: nat,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
=> ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).
% floor_eq3
tff(fact_9420_le__nat__floor,axiom,
! [Xb: nat,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).
% le_nat_floor
tff(fact_9421_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_9422_sgn__power__injE,axiom,
! [A: real,Nb: nat,Xb: 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)),Nb)) = Xb )
=> ( ( Xb = 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)),Nb)) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( A = B ) ) ) ) ).
% sgn_power_injE
tff(fact_9423_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
=> ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).
% Suc_nat_eq_nat_zadd1
tff(fact_9424_nat__less__iff,axiom,
! [W: int,Mb: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),Mb)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).
% nat_less_iff
tff(fact_9425_nat__mult__distrib__neg,axiom,
! [Z: int,Z5: int] :
( aa_int_o(aa_int_fun_int_o(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_9426_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
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)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).
% nat_abs_int_diff
tff(fact_9427_floor__eq4,axiom,
! [Nb: nat,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
=> ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).
% floor_eq4
tff(fact_9428_diff__nat__eq__if,axiom,
! [Z: int,Z5: int] :
aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
aa_int_nat(nat2,Z),
$let(
d: int,
d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
$ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).
% diff_nat_eq_if
tff(fact_9429_bit__nat__def,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
<=> ~ aa_nat_o(aa_nat_fun_nat_o(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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).
% bit_nat_def
tff(fact_9430_floor__real__def,axiom,
! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aTP_Lamm_ra(Xb)) ).
% floor_real_def
tff(fact_9431_even__nat__iff,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).
% even_nat_iff
tff(fact_9432_powr__real__of__int,axiom,
! [Xb: real,Nb: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).
% powr_real_of_int
tff(fact_9433_arctan__inverse,axiom,
! [Xb: real] :
( ( Xb != zero_zero_real )
=> ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb)) = 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,Xb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).
% arctan_inverse
tff(fact_9434_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_9435_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_9436_setceilmax,axiom,
! [S2: vEBT_VEBT,Mb: nat,Listy: list_VEBT_VEBT,Nb: nat] :
( aa_nat_o(vEBT_invar_vebt(S2),Mb)
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Listy))
=> aa_nat_o(vEBT_invar_vebt(X4),Nb) )
=> ( ( Mb = aa_nat_nat(suc,Nb) )
=> ( ! [X4: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(Listy))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X4)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) )
=> ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) )
=> ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S2,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ) ) ) ) ) ).
% setceilmax
tff(fact_9437_height__compose__list,axiom,
! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ) ).
% height_compose_list
tff(fact_9438_max__ins__scaled,axiom,
! [Nb: nat,X14: vEBT_VEBT,Mb: nat,X13: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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_9439_complex__i__mult__minus,axiom,
! [Xb: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),Xb)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).
% complex_i_mult_minus
tff(fact_9440_height__i__max,axiom,
! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
=> aa_nat_o(aa_nat_fun_nat_o(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_9441_max__idx__list,axiom,
! [I: nat,X13: list_VEBT_VEBT,Nb: nat,X14: vEBT_VEBT] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Nb),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_9442_divide__i,axiom,
! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xb) ).
% divide_i
tff(fact_9443_i__squared,axiom,
aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% i_squared
tff(fact_9444_divide__numeral__i,axiom,
! [Z: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).
% divide_numeral_i
tff(fact_9445_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_9446_exp__pi__i,axiom,
aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% exp_pi_i
tff(fact_9447_exp__pi__i_H,axiom,
aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% exp_pi_i'
tff(fact_9448_i__even__power,axiom,
! [Nb: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).
% i_even_power
tff(fact_9449_i__times__eq__iff,axiom,
! [W: complex,Z: complex] :
( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
<=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).
% i_times_eq_iff
tff(fact_9450_VEBT__internal_Oheight_Osimps_I2_J,axiom,
! [Uu2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Uu2,Dega,TreeLista,Summarya)) = 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(Summarya,set_VEBT_VEBT2(TreeLista))))) ).
% VEBT_internal.height.simps(2)
tff(fact_9451_Complex__mult__i,axiom,
! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).
% Complex_mult_i
tff(fact_9452_i__mult__Complex,axiom,
! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).
% i_mult_Complex
tff(fact_9453_Complex__eq,axiom,
! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).
% Complex_eq
tff(fact_9454_VEBT__internal_Oheight_Oelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y )
=> ( ( ? [A4: $o,B4: $o] : Xb = vEBT_Leaf((A4),(B4))
=> ( Y != zero_zero_nat ) )
=> ~ ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uu,Deg,TreeList,Summary) )
=> ( 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(Summary,set_VEBT_VEBT2(TreeList))))) ) ) ) ) ).
% VEBT_internal.height.elims
tff(fact_9455_divide__nat__def,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rb(Mb),Nb)))) ).
% divide_nat_def
tff(fact_9456_i__complex__of__real,axiom,
! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).
% i_complex_of_real
tff(fact_9457_complex__of__real__i,axiom,
! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).
% complex_of_real_i
tff(fact_9458_complex__split__polar,axiom,
! [Z: complex] :
? [R3: real,A4: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A4))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A4))))) ).
% complex_split_polar
tff(fact_9459_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_9460_cmod__complex__polar,axiom,
! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),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)))))) = aa_real_real(abs_abs_real,R2) ).
% cmod_complex_polar
tff(fact_9461_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% csqrt_ii
tff(fact_9462_floor__rat__def,axiom,
! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_rc(Xb)) ).
% floor_rat_def
tff(fact_9463_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% Arg_minus_ii
tff(fact_9464_Arg__ii,axiom,
aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).
% Arg_ii
tff(fact_9465_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_9466_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_9467_sgn__rat__def,axiom,
! [A: rat] :
aa_rat_rat(sgn_sgn_rat,A) = $ite(
A = zero_zero_rat,
zero_zero_rat,
$ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).
% sgn_rat_def
tff(fact_9468_less__eq__rat__def,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
| ( Xb = Y ) ) ) ).
% less_eq_rat_def
tff(fact_9469_obtain__pos__sum,axiom,
! [R2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
=> ~ ! [S: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S)
=> ! [T2: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T2)
=> ( R2 != aa_rat_rat(plus_plus_rat(S),T2) ) ) ) ) ).
% obtain_pos_sum
tff(fact_9470_abs__rat__def,axiom,
! [A: rat] :
aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).
% abs_rat_def
tff(fact_9471_zero__notin__Suc__image,axiom,
! [A3: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A3)) ).
% zero_notin_Suc_image
tff(fact_9472_image__Suc__lessThan,axiom,
! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb) ).
% image_Suc_lessThan
tff(fact_9473_image__Suc__atMost,axiom,
! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Nb)) ).
% image_Suc_atMost
tff(fact_9474_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) ).
% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9475_lessThan__Suc__eq__insert__0,axiom,
! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).
% lessThan_Suc_eq_insert_0
tff(fact_9476_atMost__Suc__eq__insert__0,axiom,
! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).
% atMost_Suc_eq_insert_0
tff(fact_9477_Arg__bounded,axiom,
! [Z: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).
% Arg_bounded
tff(fact_9478_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).
% cis_minus_pi_half
tff(fact_9479_rat__inverse__code,axiom,
! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_rd),quotient_of(P2)) ).
% rat_inverse_code
tff(fact_9480_cis__pi__half,axiom,
aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).
% cis_pi_half
tff(fact_9481_cis__2pi,axiom,
aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).
% cis_2pi
tff(fact_9482_divide__rat__def,axiom,
! [Q3: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q3),R2) = aa_rat_rat(times_times_rat(Q3),aa_rat_rat(inverse_inverse_rat,R2)) ).
% divide_rat_def
tff(fact_9483_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_9484_quotient__of__denom__pos,axiom,
! [R2: rat,P2: int,Q3: int] :
( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).
% quotient_of_denom_pos
tff(fact_9485_DeMoivre,axiom,
! [A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).
% DeMoivre
tff(fact_9486_cis__conv__exp,axiom,
! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).
% cis_conv_exp
tff(fact_9487_rat__less__code,axiom,
! [P2: rat,Q3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q3)
<=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_rf(Q3)),quotient_of(P2)) ) ).
% rat_less_code
tff(fact_9488_rat__less__eq__code,axiom,
! [P2: rat,Q3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q3)
<=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_rh(Q3)),quotient_of(P2)) ) ).
% rat_less_eq_code
tff(fact_9489_cis__Arg__unique,axiom,
! [Z: complex,Xb: real] :
( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xb) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
=> ( aa_complex_real(arg,Z) = Xb ) ) ) ) ).
% cis_Arg_unique
tff(fact_9490_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)) )
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).
% Arg_correct
tff(fact_9491_bij__betw__roots__unity,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_ri(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aTP_Lamm_bp(Nb))) ) ).
% bij_betw_roots_unity
tff(fact_9492_rat__plus__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_rk(Q3)),quotient_of(P2)) ).
% rat_plus_code
tff(fact_9493_Arg__def,axiom,
! [Z: complex] :
aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_rl(Z))) ).
% Arg_def
tff(fact_9494_rat__minus__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_rn(Q3)),quotient_of(P2)) ).
% rat_minus_code
tff(fact_9495_normalize__negative,axiom,
! [Q3: int,P2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Q3),zero_zero_int)
=> ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = 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,Q3))) ) ) ).
% normalize_negative
tff(fact_9496_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P2: int,Q3: int] :
( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).
% normalize_denom_pos
tff(fact_9497_normalize__crossproduct,axiom,
! [Q3: int,S2: int,P2: int,R2: int] :
( ( Q3 != zero_zero_int )
=> ( ( S2 != zero_zero_int )
=> ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S2)) )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).
% normalize_crossproduct
tff(fact_9498_rat__divide__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_rp(Q3)),quotient_of(P2)) ).
% rat_divide_code
tff(fact_9499_rat__times__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_rr(Q3)),quotient_of(P2)) ).
% rat_times_code
tff(fact_9500_cis__multiple__2pi,axiom,
! [Nb: real] :
( aa_set_real_o(member_real(Nb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).
% cis_multiple_2pi
tff(fact_9501_xor__Suc__0__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),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,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).
% xor_Suc_0_eq
tff(fact_9502_Suc__0__xor__eq,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).
% Suc_0_xor_eq
tff(fact_9503_horner__sum__of__bool__2__less,axiom,
! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,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_o(Bs))) ).
% horner_sum_of_bool_2_less
tff(fact_9504_bij__betw__Suc,axiom,
! [M6: set_nat,N4: set_nat] :
( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N4)
<=> ( image_nat_nat(suc,M6) = N4 ) ) ).
% bij_betw_Suc
tff(fact_9505_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_9506_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_9507_xor__nat__numerals_I3_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).
% xor_nat_numerals(3)
tff(fact_9508_xor__nat__numerals_I4_J,axiom,
! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).
% xor_nat_numerals(4)
tff(fact_9509_sin__times__pi__eq__0,axiom,
! [Xb: real] :
( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),pi)) = zero_zero_real )
<=> aa_set_real_o(member_real(Xb),ring_1_Ints_real) ) ).
% sin_times_pi_eq_0
tff(fact_9510_sin__integer__2pi,axiom,
! [Nb: real] :
( aa_set_real_o(member_real(Nb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).
% sin_integer_2pi
tff(fact_9511_cos__integer__2pi,axiom,
! [Nb: real] :
( aa_set_real_o(member_real(Nb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).
% cos_integer_2pi
tff(fact_9512_xor__nat__unfold,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
Mb = zero_zero_nat,
Nb,
$ite(Nb = zero_zero_nat,Mb,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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).
% xor_nat_unfold
tff(fact_9513_xor__nat__rec,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% xor_nat_rec
tff(fact_9514_bij__betw__nth__root__unity,axiom,
! [C: complex,Nb: nat] :
( ( C != zero_zero_complex )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_set_complex_o(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,Nb),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,Nb))))),collect_complex(aTP_Lamm_bp(Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_rs(C),Nb))) ) ) ).
% bij_betw_nth_root_unity
tff(fact_9515_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int)
<=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).
% xor_negative_int_iff
tff(fact_9516_push__bit__negative__int__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Nb),K)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% push_bit_negative_int_iff
tff(fact_9517_real__root__Suc__0,axiom,
! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).
% real_root_Suc_0
tff(fact_9518_real__root__eq__iff,axiom,
! [Nb: nat,Xb: real,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
<=> ( Xb = Y ) ) ) ).
% real_root_eq_iff
tff(fact_9519_real__root__eq__0__iff,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
<=> ( Xb = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
tff(fact_9520_real__root__less__iff,axiom,
! [Nb: nat,Xb: real,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).
% real_root_less_iff
tff(fact_9521_real__root__le__iff,axiom,
! [Nb: nat,Xb: real,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).
% real_root_le_iff
tff(fact_9522_real__root__eq__1__iff,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
<=> ( Xb = one_one_real ) ) ) ).
% real_root_eq_1_iff
tff(fact_9523_real__root__one,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).
% real_root_one
tff(fact_9524_real__root__lt__0__iff,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).
% real_root_lt_0_iff
tff(fact_9525_real__root__gt__0__iff,axiom,
! [Nb: nat,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).
% real_root_gt_0_iff
tff(fact_9526_real__root__le__0__iff,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).
% real_root_le_0_iff
tff(fact_9527_real__root__ge__0__iff,axiom,
! [Nb: nat,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).
% real_root_ge_0_iff
tff(fact_9528_real__root__lt__1__iff,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).
% real_root_lt_1_iff
tff(fact_9529_real__root__gt__1__iff,axiom,
! [Nb: nat,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).
% real_root_gt_1_iff
tff(fact_9530_real__root__ge__1__iff,axiom,
! [Nb: nat,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).
% real_root_ge_1_iff
tff(fact_9531_real__root__le__1__iff,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).
% real_root_le_1_iff
tff(fact_9532_push__bit__of__Suc__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),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))),Nb) ).
% push_bit_of_Suc_0
tff(fact_9533_real__root__pow__pos2,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).
% real_root_pow_pos2
tff(fact_9534_real__root__mult,axiom,
! [Nb: nat,Xb: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).
% real_root_mult
tff(fact_9535_real__root__mult__exp,axiom,
! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Xb) = aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).
% real_root_mult_exp
tff(fact_9536_set__bit__nat__def,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).
% set_bit_nat_def
tff(fact_9537_flip__bit__nat__def,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).
% flip_bit_nat_def
tff(fact_9538_real__root__less__mono,axiom,
! [Nb: nat,Xb: real,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).
% real_root_less_mono
tff(fact_9539_real__root__le__mono,axiom,
! [Nb: nat,Xb: real,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).
% real_root_le_mono
tff(fact_9540_real__root__power,axiom,
! [Nb: nat,Xb: real,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K) ) ) ).
% real_root_power
tff(fact_9541_real__root__abs,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).
% real_root_abs
tff(fact_9542_sgn__root,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).
% sgn_root
tff(fact_9543_bit__push__bit__iff__int,axiom,
! [Mb: nat,K: int,Nb: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Mb),K)),Nb)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).
% bit_push_bit_iff_int
tff(fact_9544_bit__push__bit__iff__nat,axiom,
! [Mb: nat,Q3: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q3)),Nb)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
& aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).
% bit_push_bit_iff_nat
tff(fact_9545_real__root__gt__zero,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).
% real_root_gt_zero
tff(fact_9546_real__root__strict__decreasing,axiom,
! [Nb: nat,N4: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).
% real_root_strict_decreasing
tff(fact_9547_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_9548_root__abs__power,axiom,
! [Nb: nat,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).
% root_abs_power
tff(fact_9549_real__root__pos__pos,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).
% real_root_pos_pos
tff(fact_9550_real__root__strict__increasing,axiom,
! [Nb: nat,N4: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N4),Xb)) ) ) ) ) ).
% real_root_strict_increasing
tff(fact_9551_real__root__decreasing,axiom,
! [Nb: nat,N4: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).
% real_root_decreasing
tff(fact_9552_real__root__pow__pos,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).
% real_root_pow_pos
tff(fact_9553_real__root__power__cancel,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).
% real_root_power_cancel
tff(fact_9554_real__root__pos__unique,axiom,
! [Nb: nat,Y: real,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ) ).
% real_root_pos_unique
tff(fact_9555_odd__real__root__power__cancel,axiom,
! [Nb: nat,Xb: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ).
% odd_real_root_power_cancel
tff(fact_9556_odd__real__root__unique,axiom,
! [Nb: nat,Y: real,Xb: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ).
% odd_real_root_unique
tff(fact_9557_odd__real__root__pow,axiom,
! [Nb: nat,Xb: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ).
% odd_real_root_pow
tff(fact_9558_push__bit__int__def,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),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))),Nb)) ).
% push_bit_int_def
tff(fact_9559_push__bit__nat__def,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% push_bit_nat_def
tff(fact_9560_real__root__increasing,axiom,
! [Nb: nat,N4: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N4),Xb)) ) ) ) ) ).
% real_root_increasing
tff(fact_9561_sgn__power__root,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),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,Nb),Xb))),Nb)) = Xb ) ) ).
% sgn_power_root
tff(fact_9562_root__sgn__power,axiom,
! [Nb: nat,Y: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),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)),Nb))) = Y ) ) ).
% root_sgn_power
tff(fact_9563_push__bit__minus__one,axiom,
! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),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))),Nb)) ).
% push_bit_minus_one
tff(fact_9564_log__root,axiom,
! [Nb: nat,A: real,B: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).
% log_root
tff(fact_9565_log__base__root,axiom,
! [Nb: nat,B: real,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).
% log_base_root
tff(fact_9566_ln__root,axiom,
! [Nb: nat,B: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).
% ln_root
tff(fact_9567_XOR__upper,axiom,
! [Xb: int,Nb: nat,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).
% XOR_upper
tff(fact_9568_split__root,axiom,
! [P: fun_real_o,Nb: nat,Xb: real] :
( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
<=> ( ( ( Nb = zero_zero_nat )
=> aa_real_o(P,zero_zero_real) )
& ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ! [Y5: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),Nb)) = Xb )
=> aa_real_o(P,Y5) ) ) ) ) ).
% split_root
tff(fact_9569_root__powr__inverse,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).
% root_powr_inverse
tff(fact_9570_xor__int__rec,axiom,
! [K: int,L: int] :
aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(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(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% xor_int_rec
tff(fact_9571_xor__int__unfold,axiom,
! [K: int,L: int] :
aa_int_int(bit_se6526347334894502574or_int(K),L) = $ite(
K = aa_int_int(uminus_uminus_int,one_one_int),
aa_int_int(bit_ri7919022796975470100ot_int,L),
$ite(
L = aa_int_int(uminus_uminus_int,one_one_int),
aa_int_int(bit_ri7919022796975470100ot_int,K),
$ite(
K = zero_zero_int,
L,
$ite(L = zero_zero_int,K,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(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_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(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).
% xor_int_unfold
tff(fact_9572_VEBT__internal_Oheight_Opelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xb)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = zero_zero_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B4))) ) )
=> ~ ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Uu,Deg,TreeList,Summary) )
=> ( ( 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(Summary,set_VEBT_VEBT2(TreeList))))) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Node(Uu,Deg,TreeList,Summary)) ) ) ) ) ) ).
% VEBT_internal.height.pelims
tff(fact_9573_Cauchy__iff2,axiom,
! [X6: fun_nat_real] :
( topolo4055970368930404560y_real(X6)
<=> ! [J3: nat] :
? [M7: nat] :
! [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M4)
=> ! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N3)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).
% Cauchy_iff2
tff(fact_9574_Sum__Ico__nat,axiom,
! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = 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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Sum_Ico_nat
tff(fact_9575_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_9576_not__negative__int__iff,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).
% not_negative_int_iff
tff(fact_9577_not__nonnegative__int__iff,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% not_nonnegative_int_iff
tff(fact_9578_atLeastLessThan__singleton,axiom,
! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).
% atLeastLessThan_singleton
tff(fact_9579_all__nat__less__eq,axiom,
! [Nb: nat,P: fun_nat_o] :
( ! [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
=> aa_nat_o(P,M4) )
<=> ! [X: nat] :
( aa_set_nat_o(member_nat(X),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
=> aa_nat_o(P,X) ) ) ).
% all_nat_less_eq
tff(fact_9580_ex__nat__less__eq,axiom,
! [Nb: nat,P: fun_nat_o] :
( ? [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
& aa_nat_o(P,M4) )
<=> ? [X: nat] :
( aa_set_nat_o(member_nat(X),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
& aa_nat_o(P,X) ) ) ).
% ex_nat_less_eq
tff(fact_9581_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_9582_atLeast0__lessThan__Suc,axiom,
! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) ).
% atLeast0_lessThan_Suc
tff(fact_9583_not__int__div__2,axiom,
! [K: int] : aa_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(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% not_int_div_2
tff(fact_9584_even__not__iff__int,axiom,
! [K: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K))
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).
% even_not_iff_int
tff(fact_9585_and__not__numerals_I2_J,axiom,
! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).
% and_not_numerals(2)
tff(fact_9586_and__not__numerals_I4_J,axiom,
! [Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).
% and_not_numerals(4)
tff(fact_9587_or__not__numerals_I2_J,axiom,
! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).
% or_not_numerals(2)
tff(fact_9588_or__not__numerals_I4_J,axiom,
! [Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).
% or_not_numerals(4)
tff(fact_9589_atLeastLessThanSuc,axiom,
! [Mb: nat,Nb: nat] :
aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)),bot_bot_set_nat) ).
% atLeastLessThanSuc
tff(fact_9590_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))) ).
% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9591_prod__Suc__fact,axiom,
! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).
% prod_Suc_fact
tff(fact_9592_prod__Suc__Suc__fact,axiom,
! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).
% prod_Suc_Suc_fact
tff(fact_9593_and__not__numerals_I5_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).
% and_not_numerals(5)
tff(fact_9594_and__not__numerals_I7_J,axiom,
! [Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).
% and_not_numerals(7)
tff(fact_9595_or__not__numerals_I3_J,axiom,
! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).
% or_not_numerals(3)
tff(fact_9596_atLeastLessThan__nat__numeral,axiom,
! [Mb: nat,K: num] :
aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).
% atLeastLessThan_nat_numeral
tff(fact_9597_and__not__numerals_I6_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).
% and_not_numerals(6)
tff(fact_9598_and__not__numerals_I9_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).
% and_not_numerals(9)
tff(fact_9599_or__not__numerals_I6_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).
% or_not_numerals(6)
tff(fact_9600_or__not__numerals_I5_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).
% or_not_numerals(5)
tff(fact_9601_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Xb: nat,Y: nat] :
image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xb),Y)) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_9602_atLeast1__lessThan__eq__remove0,axiom,
! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).
% atLeast1_lessThan_eq_remove0
tff(fact_9603_and__not__numerals_I8_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).
% and_not_numerals(8)
tff(fact_9604_or__not__numerals_I8_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).
% or_not_numerals(8)
tff(fact_9605_or__not__numerals_I9_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).
% or_not_numerals(9)
tff(fact_9606_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_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(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(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% not_int_rec
tff(fact_9607_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_9608_vebt__maxt_Opelims,axiom,
! [Xb: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_maxt(Xb) = Y )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = $ite(
(B4),
some_nat(one_one_nat),
$ite((A4),some_nat(zero_zero_nat),none_nat) ) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A4),(B4))) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ( Y = none_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
=> ( ( Y = some_nat(Ma) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).
% vebt_maxt.pelims
tff(fact_9609_vebt__mint_Opelims,axiom,
! [Xb: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_mint(Xb) = Y )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = $ite(
(A4),
some_nat(zero_zero_nat),
$ite((B4),some_nat(one_one_nat),none_nat) ) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B4))) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ( Y = none_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
=> ( ( Y = some_nat(Mi) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).
% vebt_mint.pelims
tff(fact_9610_Chebyshev__sum__upper__nat,axiom,
! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
( ! [I3: nat,J2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2)) ) )
=> ( ! [I3: nat,J2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I3)) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nf(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),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),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).
% Chebyshev_sum_upper_nat
tff(fact_9611_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xb)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((A4),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf((A4),(B4))) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9612_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xb)
=> ( ! [A4: $o,B4: $o] :
( ( Xb = vEBT_Leaf((A4),(B4)) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
$ite((B4),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf((A4),(B4))) ) )
=> ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
=> ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9613_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
! [Xb: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xb)
=> ( ( ( Xb = vEBT_Leaf($false,$false) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
=> ( ! [Uv: $o] :
( ( Xb = vEBT_Leaf($true,(Uv)) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv))) ) )
=> ( ! [Uu: $o] :
( ( Xb = vEBT_Leaf((Uu),$true) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu),$true)) ) )
=> ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
=> ( ( Y = one_one_nat )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9614_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Y: $o] :
( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
<=> (Y) )
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
=> ( ( ( Xb = vEBT_Leaf($false,$false) )
=> ( (Y)
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
=> ( ! [Uv: $o] :
( ( Xb = vEBT_Leaf($true,(Uv)) )
=> ( ~ (Y)
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) ) )
=> ( ! [Uu: $o] :
( ( Xb = vEBT_Leaf((Uu),$true) )
=> ( ~ (Y)
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) ) )
=> ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
=> ( (Y)
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
=> ( ~ (Y)
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
tff(fact_9615_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [Xb: vEBT_VEBT] :
( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
=> ( ! [Uv: $o] :
( ( Xb = vEBT_Leaf($true,(Uv)) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) )
=> ( ! [Uu: $o] :
( ( Xb = vEBT_Leaf((Uu),$true) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
tff(fact_9616_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [Xb: vEBT_VEBT] :
( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
=> ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
=> ( ( ( Xb = vEBT_Leaf($false,$false) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
=> ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
=> ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
tff(fact_9617_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_9618_valid__eq,axiom,
! [Ta: vEBT_VEBT,D: nat] :
( aa_nat_o(vEBT_VEBT_valid(Ta),D)
<=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).
% valid_eq
tff(fact_9619_valid__eq1,axiom,
! [Ta: vEBT_VEBT,D: nat] :
( aa_nat_o(vEBT_invar_vebt(Ta),D)
=> aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).
% valid_eq1
tff(fact_9620_valid__eq2,axiom,
! [Ta: vEBT_VEBT,D: nat] :
( aa_nat_o(vEBT_VEBT_valid(Ta),D)
=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).
% valid_eq2
tff(fact_9621_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,D: nat] :
( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu2),(Uv2))),D)
<=> ( D = one_one_nat ) ) ).
% VEBT_internal.valid'.simps(1)
tff(fact_9622_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_9623_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% csqrt.simps(1)
tff(fact_9624_divmod__step__integer__def,axiom,
! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ed(L),Qr) ).
% divmod_step_integer_def
tff(fact_9625_sgn__integer__code,axiom,
! [K: code_integer] :
aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
K = zero_z3403309356797280102nteger,
zero_z3403309356797280102nteger,
$ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).
% sgn_integer_code
tff(fact_9626_times__integer__code_I2_J,axiom,
! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).
% times_integer_code(2)
tff(fact_9627_times__integer__code_I1_J,axiom,
! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% times_integer_code(1)
tff(fact_9628_scaleR__complex_Osimps_I1_J,axiom,
! [R2: real,Xb: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb)) ).
% scaleR_complex.simps(1)
tff(fact_9629_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
tff(fact_9630_cos__n__Re__cis__pow__n,axiom,
! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).
% cos_n_Re_cis_pow_n
tff(fact_9631_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).
% VEBT.size_gen(2)
tff(fact_9632_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
$ite(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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).
% csqrt.code
tff(fact_9633_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,
$ite(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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).
% csqrt.simps(2)
tff(fact_9634_integer__of__int__code,axiom,
! [K: int] :
aa_int_Code_integer(code_integer_of_int,K) = $ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
$ite(
K = zero_zero_int,
zero_z3403309356797280102nteger,
$let(
l: code_integer,
l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
$ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).
% integer_of_int_code
tff(fact_9635_csqrt__of__real__nonpos,axiom,
! [Xb: complex] :
( ( aa_complex_real(im,Xb) = zero_zero_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
=> ( aa_complex_complex(csqrt,Xb) = 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,Xb))))) ) ) ) ).
% csqrt_of_real_nonpos
tff(fact_9636_Im__i__times,axiom,
! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_complex_real(re,Z) ).
% Im_i_times
tff(fact_9637_Re__i__times,axiom,
! [Z: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).
% Re_i_times
tff(fact_9638_csqrt__minus,axiom,
! [Xb: complex] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
| ( ( aa_complex_real(im,Xb) = zero_zero_real )
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
=> ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).
% csqrt_minus
tff(fact_9639_less__integer_Oabs__eq,axiom,
! [Xaa: int,Xb: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xaa),Xb) ) ).
% less_integer.abs_eq
tff(fact_9640_abs__integer__code,axiom,
! [K: code_integer] :
abs_abs_Code_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).
% abs_integer_code
tff(fact_9641_less__integer__code_I1_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).
% less_integer_code(1)
tff(fact_9642_times__integer_Oabs__eq,axiom,
! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xaa),Xb)) ).
% times_integer.abs_eq
tff(fact_9643_scaleR__complex_Osimps_I2_J,axiom,
! [R2: real,Xb: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb)) ).
% scaleR_complex.simps(2)
tff(fact_9644_times__complex_Osimps_I2_J,axiom,
! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))) ).
% times_complex.simps(2)
tff(fact_9645_times__complex_Osimps_I1_J,axiom,
! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).
% times_complex.simps(1)
tff(fact_9646_scaleR__complex_Ocode,axiom,
! [R2: real,Xb: complex] : aa_complex_complex(real_V2046097035970521341omplex(R2),Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb))) ).
% scaleR_complex.code
tff(fact_9647_csqrt__principal,axiom,
! [Z: complex] :
( aa_real_o(aa_real_fun_real_o(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 )
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).
% csqrt_principal
tff(fact_9648_sin__n__Im__cis__pow__n,axiom,
! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).
% sin_n_Im_cis_pow_n
tff(fact_9649_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_9650_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_9651_complex__eq,axiom,
! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).
% complex_eq
tff(fact_9652_times__complex_Ocode,axiom,
! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(Xb),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),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,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y)))) ).
% times_complex.code
tff(fact_9653_exp__eq__polar,axiom,
! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).
% exp_eq_polar
tff(fact_9654_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_9655_Im__power2,axiom,
! [Xb: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xb),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Xb))),aa_complex_real(im,Xb)) ).
% Im_power2
tff(fact_9656_Re__power2,axiom,
! [Xb: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xb),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,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% Re_power2
tff(fact_9657_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_9658_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_9659_inverse__complex_Osimps_I1_J,axiom,
! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% inverse_complex.simps(1)
tff(fact_9660_complex__neq__0,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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_9661_Re__divide,axiom,
! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xb),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),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_9662_csqrt__square,axiom,
! [B: complex] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
| ( ( aa_complex_real(re,B) = zero_zero_real )
& aa_real_o(aa_real_fun_real_o(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_9663_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 )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
| ( ( aa_complex_real(re,W) = zero_zero_real )
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
=> ( aa_complex_complex(csqrt,Z) = W ) ) ) ).
% csqrt_unique
tff(fact_9664_inverse__complex_Osimps_I2_J,axiom,
! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% inverse_complex.simps(2)
tff(fact_9665_Im__divide,axiom,
! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xb),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),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_9666_complex__abs__le__norm,axiom,
! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).
% complex_abs_le_norm
tff(fact_9667_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_9668_inverse__complex_Ocode,axiom,
! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),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,Xb)),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,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% inverse_complex.code
tff(fact_9669_Complex__divide,axiom,
! [Xb: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),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_9670_Im__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
=> ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),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,R2))),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_9671_Re__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
=> ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R2)),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_9672_real__eq__imaginary__iff,axiom,
! [Y: complex,Xb: complex] :
( aa_set_complex_o(member_complex(Y),real_V2521375963428798218omplex)
=> ( aa_set_complex_o(member_complex(Xb),real_V2521375963428798218omplex)
=> ( ( Xb = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
<=> ( ( Xb = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% real_eq_imaginary_iff
tff(fact_9673_imaginary__eq__real__iff,axiom,
! [Y: complex,Xb: complex] :
( aa_set_complex_o(member_complex(Y),real_V2521375963428798218omplex)
=> ( aa_set_complex_o(member_complex(Xb),real_V2521375963428798218omplex)
=> ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = Xb )
<=> ( ( Xb = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% imaginary_eq_real_iff
tff(fact_9674_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_9675_int__of__integer__code,axiom,
! [K: code_integer] :
code_int_of_integer(K) = $ite(
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
$ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_ru,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).
% int_of_integer_code
tff(fact_9676_num__of__integer__code,axiom,
! [K: code_integer] :
code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_rv,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% num_of_integer_code
tff(fact_9677_complex__cnj__mult,axiom,
! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).
% complex_cnj_mult
tff(fact_9678_times__integer_Orep__eq,axiom,
! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).
% times_integer.rep_eq
tff(fact_9679_complex__In__mult__cnj__zero,axiom,
! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).
% complex_In_mult_cnj_zero
tff(fact_9680_less__integer_Orep__eq,axiom,
! [Xb: code_integer,Xaa: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Xaa)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).
% less_integer.rep_eq
tff(fact_9681_integer__less__iff,axiom,
! [K: code_integer,L: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).
% integer_less_iff
tff(fact_9682_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_9683_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_9684_complex__mod__sqrt__Re__mult__cnj,axiom,
! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)))) ).
% complex_mod_sqrt_Re_mult_cnj
tff(fact_9685_Re__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(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_9686_Re__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
<=> aa_real_o(aa_real_fun_real_o(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_9687_Re__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
<=> aa_real_o(aa_real_fun_real_o(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_9688_Re__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(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_9689_Im__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(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_9690_Im__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
<=> aa_real_o(aa_real_fun_real_o(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_9691_Im__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
<=> aa_real_o(aa_real_fun_real_o(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_9692_Im__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
<=> aa_real_o(aa_real_fun_real_o(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_9693_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_9694_complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
& ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
<=> aa_real_o(aa_real_fun_real_o(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_9695_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_9696_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).
% complex_add_cnj
tff(fact_9697_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).
% complex_diff_cnj
tff(fact_9698_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_9699_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,aa_num_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_9700_nat__of__integer__code,axiom,
! [K: code_integer] :
code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_rw,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% nat_of_integer_code
tff(fact_9701_card__Collect__less__nat,axiom,
! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Nb))) = Nb ).
% card_Collect_less_nat
tff(fact_9702_card__atMost,axiom,
! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).
% card_atMost
tff(fact_9703_card__Collect__le__nat,axiom,
! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bi,Nb))) = aa_nat_nat(suc,Nb) ).
% card_Collect_le_nat
tff(fact_9704_card__atLeastAtMost,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).
% card_atLeastAtMost
tff(fact_9705_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
<=> aa_nat_o(case_nat_o($false,aTP_Lamm_rx),Nat) ) ).
% nat.disc_eq_case(2)
tff(fact_9706_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
<=> aa_nat_o(case_nat_o($true,aTP_Lamm_ry),Nat) ) ).
% nat.disc_eq_case(1)
tff(fact_9707_card__less,axiom,
! [M6: set_nat,I: nat] :
( aa_set_nat_o(member_nat(zero_zero_nat),M6)
=> ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rz(M6),I))) != zero_zero_nat ) ) ).
% card_less
tff(fact_9708_card__less__Suc,axiom,
! [M6: set_nat,I: nat] :
( aa_set_nat_o(member_nat(zero_zero_nat),M6)
=> ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sa(M6),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rz(M6),I))) ) ) ).
% card_less_Suc
tff(fact_9709_card__less__Suc2,axiom,
! [M6: set_nat,I: nat] :
( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
=> ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sa(M6),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rz(M6),I))) ) ) ).
% card_less_Suc2
tff(fact_9710_subset__card__intvl__is__intvl,axiom,
! [A3: set_nat,K: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(A3),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A3))))
=> ( A3 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A3))) ) ) ).
% subset_card_intvl_is_intvl
tff(fact_9711_less__eq__nat_Osimps_I2_J,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
<=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).
% less_eq_nat.simps(2)
tff(fact_9712_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_9713_max__Suc2,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_sb,Nb)),Mb) ).
% max_Suc2
tff(fact_9714_max__Suc1,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_sc,Nb)),Mb) ).
% max_Suc1
tff(fact_9715_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_9716_card__le__Suc__Max,axiom,
! [S3: set_nat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(S3)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S3))) ) ).
% card_le_Suc_Max
tff(fact_9717_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set_nat,Nb: nat] :
( aa_set_nat_o(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N4)),Nb) ) ).
% subset_eq_atLeast0_lessThan_card
tff(fact_9718_card__sum__le__nat__sum,axiom,
! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),S3)) ).
% card_sum_le_nat_sum
tff(fact_9719_card__nth__roots,axiom,
! [C: complex,Nb: nat] :
( ( C != zero_zero_complex )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_rs(C),Nb))) = Nb ) ) ) ).
% card_nth_roots
tff(fact_9720_card__roots__unity__eq,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( finite_card_complex(collect_complex(aTP_Lamm_bp(Nb))) = Nb ) ) ).
% card_roots_unity_eq
tff(fact_9721_diff__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_aj),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).
% diff_Suc
tff(fact_9722_pred__def,axiom,
! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_aj),Nat) ).
% pred_def
tff(fact_9723_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,aa_num_num(inc,K)))) ).
% drop_bit_numeral_minus_bit1
tff(fact_9724_drop__bit__negative__int__iff,axiom,
! [Nb: nat,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Nb),K)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).
% drop_bit_negative_int_iff
tff(fact_9725_drop__bit__Suc__minus__bit0,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).
% drop_bit_Suc_minus_bit0
tff(fact_9726_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_9727_drop__bit__Suc__minus__bit1,axiom,
! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).
% drop_bit_Suc_minus_bit1
tff(fact_9728_drop__bit__int__def,axiom,
! [Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K) = aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).
% drop_bit_int_def
tff(fact_9729_prod__decode__aux_Oelims,axiom,
! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
( ( nat_prod_decode_aux(Xb,Xaa) = Y )
=> ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).
% prod_decode_aux.elims
tff(fact_9730_prod__decode__aux_Osimps,axiom,
! [K: nat,Mb: nat] :
nat_prod_decode_aux(K,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K)))) ).
% prod_decode_aux.simps
tff(fact_9731_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_9732_drop__bit__of__Suc__0,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).
% drop_bit_of_Suc_0
tff(fact_9733_drop__bit__nat__def,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).
% drop_bit_nat_def
tff(fact_9734_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_9735_finite__enumerate,axiom,
! [S3: set_nat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ? [R3: fun_nat_nat] :
( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S3)))
& ! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),finite_card_nat(S3))
=> aa_set_nat_o(member_nat(aa_nat_nat(R3,N5)),S3) ) ) ) ).
% finite_enumerate
tff(fact_9736_bit__cut__integer__code,axiom,
! [K: code_integer] :
code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_sd(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% bit_cut_integer_code
tff(fact_9737_quotient__of__denom__pos_H,axiom,
! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2))) ).
% quotient_of_denom_pos'
tff(fact_9738_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K)) ).
% bit_cut_integer_def
tff(fact_9739_minus__one__mod__numeral,axiom,
! [Nb: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb))) ).
% minus_one_mod_numeral
tff(fact_9740_one__mod__minus__numeral,axiom,
! [Nb: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb)))) ).
% one_mod_minus_numeral
tff(fact_9741_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
code_divmod_integer(K,L) = $ite(
K = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
$ite(
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
$ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_se(L),code_divmod_abs(K,L))),
$ite(
L = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
$ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_sf(L),code_divmod_abs(K,L)))) ) ) ) ).
% divmod_integer_code
tff(fact_9742_bezw_Oelims,axiom,
! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
( ( bezw(Xb,Xaa) = Y )
=> ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).
% bezw.elims
tff(fact_9743_bezw_Osimps,axiom,
! [Xb: nat,Y: nat] :
bezw(Xb,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).
% bezw.simps
tff(fact_9744_bezw__non__0,axiom,
! [Y: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
=> ( bezw(Xb,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).
% bezw_non_0
tff(fact_9745_bezw_Opelims,axiom,
! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
( ( bezw(Xb,Xaa) = Y )
=> ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
=> ~ ( ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
=> ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).
% bezw.pelims
tff(fact_9746_prod__decode__aux_Opelims,axiom,
! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
( ( nat_prod_decode_aux(Xb,Xaa) = Y )
=> ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
=> ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
=> ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_9747_normalize__def,axiom,
! [P2: product_prod_int_int] :
aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P2)),
$let(
a: int,
a:= aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2)),
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(product_fst_int_int(P2)),a)),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ),
$ite(
product_snd_int_int(P2) = zero_zero_int,
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
$let(
a: int,
a:= aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2))),
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(product_fst_int_int(P2)),a)),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ) ) ) ).
% normalize_def
tff(fact_9748_gcd__pos__int,axiom,
! [Mb: int,Nb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(Mb),Nb))
<=> ( ( Mb != zero_zero_int )
| ( Nb != zero_zero_int ) ) ) ).
% gcd_pos_int
tff(fact_9749_bezout__int,axiom,
! [Xb: int,Y: int] :
? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(gcd_gcd_int(Xb),Y) ).
% bezout_int
tff(fact_9750_gcd__mult__distrib__int,axiom,
! [K: int,Mb: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(gcd_gcd_int(Mb),Nb)) = aa_int_int(gcd_gcd_int(aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb)) ).
% gcd_mult_distrib_int
tff(fact_9751_gcd__le2__int,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),B) ) ).
% gcd_le2_int
tff(fact_9752_gcd__le1__int,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),A) ) ).
% gcd_le1_int
tff(fact_9753_gcd__non__0__int,axiom,
! [Y: int,Xb: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
=> ( aa_int_int(gcd_gcd_int(Xb),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(Xb),Y)) ) ) ).
% gcd_non_0_int
tff(fact_9754_nat__descend__induct,axiom,
! [Nb: nat,P: fun_nat_o,Mb: nat] :
( ! [K3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K3)
=> aa_nat_o(P,K3) )
=> ( ! [K3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Nb)
=> ( ! [I2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),I2)
=> aa_nat_o(P,I2) )
=> aa_nat_o(P,K3) ) )
=> aa_nat_o(P,Mb) ) ) ).
% nat_descend_induct
tff(fact_9755_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
code_divmod_integer(K,L) = $ite(
K = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
$ite(
L = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
$ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_sg(L),code_divmod_abs(K,L)))) ) ) ).
% divmod_integer_eq_cases
tff(fact_9756_gcd__1__nat,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).
% gcd_1_nat
tff(fact_9757_gcd__Suc__0,axiom,
! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).
% gcd_Suc_0
tff(fact_9758_gcd__pos__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
<=> ( ( Mb != zero_zero_nat )
| ( Nb != zero_zero_nat ) ) ) ).
% gcd_pos_nat
tff(fact_9759_gcd__mult__distrib__nat,axiom,
! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).
% gcd_mult_distrib_nat
tff(fact_9760_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).
% gcd_le1_nat
tff(fact_9761_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).
% gcd_le2_nat
tff(fact_9762_gcd__diff2__nat,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).
% gcd_diff2_nat
tff(fact_9763_gcd__diff1__nat,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).
% gcd_diff1_nat
tff(fact_9764_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).
% bezout_nat
tff(fact_9765_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X4: nat,Y3: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4))
& ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4))
& ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).
% bezout_gcd_nat'
tff(fact_9766_gcd__is__Max__divisors__nat,axiom,
! [Nb: nat,Mb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sh(Nb),Mb))) ) ) ).
% gcd_is_Max_divisors_nat
tff(fact_9767_bezw__aux,axiom,
! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).
% bezw_aux
tff(fact_9768_gcd__nat_Opelims,axiom,
! [Xb: nat,Xaa: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
=> ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
=> ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
=> ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).
% gcd_nat.pelims
tff(fact_9769_xor__minus__numerals_I1_J,axiom,
! [Nb: num,K: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(aa_num_int(neg_numeral_sub_int(Nb),one)),K)) ).
% xor_minus_numerals(1)
tff(fact_9770_xor__minus__numerals_I2_J,axiom,
! [K: int,Nb: num] : aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K),aa_num_int(neg_numeral_sub_int(Nb),one))) ).
% xor_minus_numerals(2)
tff(fact_9771_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_9772_sub__BitM__One__eq,axiom,
! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).
% sub_BitM_One_eq
tff(fact_9773_Suc__funpow,axiom,
! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).
% Suc_funpow
tff(fact_9774_max__nat_Osemilattice__neutr__order__axioms,axiom,
semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bi,aTP_Lamm_bj) ).
% max_nat.semilattice_neutr_order_axioms
tff(fact_9775_times__int_Oabs__eq,axiom,
! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_sj),Xaa),Xb)) ).
% times_int.abs_eq
tff(fact_9776_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X4: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)) ).
% eq_Abs_Integ
tff(fact_9777_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_9778_int__def,axiom,
! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).
% int_def
tff(fact_9779_uminus__int_Oabs__eq,axiom,
! [Xb: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_sk),Xb)) ).
% uminus_int.abs_eq
tff(fact_9780_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_9781_less__int_Oabs__eq,axiom,
! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
<=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_sm),Xaa),Xb) ) ).
% less_int.abs_eq
tff(fact_9782_less__eq__int_Oabs__eq,axiom,
! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
<=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_so),Xaa),Xb) ) ).
% less_eq_int.abs_eq
tff(fact_9783_plus__int_Oabs__eq,axiom,
! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_sq),Xaa),Xb)) ).
% plus_int.abs_eq
tff(fact_9784_minus__int_Oabs__eq,axiom,
! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ss),Xaa),Xb)) ).
% minus_int.abs_eq
tff(fact_9785_num__of__nat_Osimps_I2_J,axiom,
! [Nb: nat] :
aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).
% num_of_nat.simps(2)
tff(fact_9786_num__of__nat__numeral__eq,axiom,
! [Q3: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q3)) = Q3 ).
% num_of_nat_numeral_eq
tff(fact_9787_num__of__nat_Osimps_I1_J,axiom,
aa_nat_num(num_of_nat,zero_zero_nat) = one ).
% num_of_nat.simps(1)
tff(fact_9788_numeral__num__of__nat,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).
% numeral_num_of_nat
tff(fact_9789_num__of__nat__One,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
=> ( aa_nat_num(num_of_nat,Nb) = one ) ) ).
% num_of_nat_One
tff(fact_9790_num__of__nat__double,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).
% num_of_nat_double
tff(fact_9791_num__of__nat__plus__distrib,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).
% num_of_nat_plus_distrib
tff(fact_9792_less__eq__int_Orep__eq,axiom,
! [Xb: int,Xaa: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa)
<=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_so),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).
% less_eq_int.rep_eq
tff(fact_9793_less__int_Orep__eq,axiom,
! [Xb: int,Xaa: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xaa)
<=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_sm),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).
% less_int.rep_eq
tff(fact_9794_uminus__int__def,axiom,
uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_sk)) ).
% uminus_int_def
tff(fact_9795_pow_Osimps_I3_J,axiom,
! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xb),Y))),Xb) ).
% pow.simps(3)
tff(fact_9796_sqr_Osimps_I2_J,axiom,
! [Nb: num] : aa_num_num(sqr,aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Nb))) ).
% sqr.simps(2)
tff(fact_9797_sqr_Osimps_I1_J,axiom,
aa_num_num(sqr,one) = one ).
% sqr.simps(1)
tff(fact_9798_sqr__conv__mult,axiom,
! [Xb: num] : aa_num_num(sqr,Xb) = aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Xb) ).
% sqr_conv_mult
tff(fact_9799_pow_Osimps_I2_J,axiom,
! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xb),Y)) ).
% pow.simps(2)
tff(fact_9800_sqr_Osimps_I3_J,axiom,
! [Nb: num] : aa_num_num(sqr,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,Nb)),Nb))) ).
% sqr.simps(3)
tff(fact_9801_times__int__def,axiom,
times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sj)) ).
% times_int_def
tff(fact_9802_minus__int__def,axiom,
minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ss)) ).
% minus_int_def
tff(fact_9803_plus__int__def,axiom,
plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sq)) ).
% plus_int_def
tff(fact_9804_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_9805_rat__floor__lemma,axiom,
! [A: int,B: int] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(fract(A),B))
& aa_rat_o(aa_rat_fun_rat_o(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(divide_divide_int(A),B)),one_one_int))) ) ).
% rat_floor_lemma
tff(fact_9806_mult__rat,axiom,
! [A: int,B: int,C: int,D: int] : aa_rat_rat(times_times_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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ).
% mult_rat
tff(fact_9807_divide__rat,axiom,
! [A: int,B: int,C: int,D: int] : aa_rat_rat(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_9808_less__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
<=> aa_int_o(aa_int_fun_int_o(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_9809_add__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( aa_rat_rat(plus_plus_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(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,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).
% add_rat
tff(fact_9810_le__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_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))) ) ) ) ).
% le_rat
tff(fact_9811_diff__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( aa_rat_rat(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_9812_sgn__rat,axiom,
! [A: int,B: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(fract(A),B)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B))) ).
% sgn_rat
tff(fact_9813_Rat__induct__pos,axiom,
! [P: fun_rat_o,Q3: rat] :
( ! [A4: int,B4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
=> aa_rat_o(P,aa_int_rat(fract(A4),B4)) )
=> aa_rat_o(P,Q3) ) ).
% Rat_induct_pos
tff(fact_9814_mult__rat__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_rat(fract(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_rat(fract(A),B) ) ) ).
% mult_rat_cancel
tff(fact_9815_eq__rat_I1_J,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( aa_int_rat(fract(A),B) = aa_int_rat(fract(C),D) )
<=> ( 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) ) ) ) ) ).
% eq_rat(1)
tff(fact_9816_zero__less__Fract__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).
% zero_less_Fract_iff
tff(fact_9817_Fract__less__zero__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).
% Fract_less_zero_iff
tff(fact_9818_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),one_one_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% Fract_less_one_iff
tff(fact_9819_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).
% one_less_Fract_iff
tff(fact_9820_integer__of__num__triv_I1_J,axiom,
code_integer_of_num(one) = one_one_Code_integer ).
% integer_of_num_triv(1)
tff(fact_9821_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).
% Fract_le_zero_iff
tff(fact_9822_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).
% zero_le_Fract_iff
tff(fact_9823_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% Fract_le_one_iff
tff(fact_9824_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% one_le_Fract_iff
tff(fact_9825_integer__of__num_I2_J,axiom,
! [Nb: num] :
code_integer_of_num(aa_num_num(bit0,Nb)) = $let(
k: code_integer,
k:= code_integer_of_num(Nb),
plus_p5714425477246183910nteger(k,k) ) ).
% integer_of_num(2)
tff(fact_9826_take__bit__numeral__minus__numeral__int,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_st(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).
% take_bit_numeral_minus_numeral_int
tff(fact_9827_positive__rat,axiom,
! [A: int,B: int] :
( aa_rat_o(positive,aa_int_rat(fract(A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).
% positive_rat
tff(fact_9828_take__bit__num__simps_I1_J,axiom,
! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).
% take_bit_num_simps(1)
tff(fact_9829_take__bit__num__simps_I2_J,axiom,
! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).
% take_bit_num_simps(2)
tff(fact_9830_take__bit__num__simps_I5_J,axiom,
! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).
% take_bit_num_simps(5)
tff(fact_9831_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_su,Nb) ).
% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9832_Rat_Opositive__mult,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(positive,Xb)
=> ( aa_rat_o(positive,Y)
=> aa_rat_o(positive,aa_rat_rat(times_times_rat(Xb),Y)) ) ) ).
% Rat.positive_mult
tff(fact_9833_less__rat__def,axiom,
! [Xb: rat,Y: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
<=> aa_rat_o(positive,aa_rat_rat(minus_minus_rat(Y),Xb)) ) ).
% less_rat_def
tff(fact_9834_take__bit__num__def,axiom,
! [Nb: nat,Mb: num] :
bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).
% take_bit_num_def
tff(fact_9835_and__minus__numerals_I3_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).
% and_minus_numerals(3)
tff(fact_9836_and__minus__numerals_I7_J,axiom,
! [Nb: num,Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).
% and_minus_numerals(7)
tff(fact_9837_and__minus__numerals_I4_J,axiom,
! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).
% and_minus_numerals(4)
tff(fact_9838_take__bit__num__simps_I4_J,axiom,
! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).
% take_bit_num_simps(4)
tff(fact_9839_take__bit__num__simps_I3_J,axiom,
! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sv,bit_take_bit_num(Nb,Mb)) ).
% take_bit_num_simps(3)
tff(fact_9840_take__bit__num__simps_I7_J,axiom,
! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).
% take_bit_num_simps(7)
tff(fact_9841_take__bit__num__simps_I6_J,axiom,
! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sv,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).
% take_bit_num_simps(6)
tff(fact_9842_and__minus__numerals_I8_J,axiom,
! [Nb: num,Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).
% and_minus_numerals(8)
tff(fact_9843_and__not__num_Osimps_I1_J,axiom,
bit_and_not_num(one,one) = none_num ).
% and_not_num.simps(1)
tff(fact_9844_and__not__num_Osimps_I8_J,axiom,
! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_sw,bit_and_not_num(Mb,Nb)) ).
% and_not_num.simps(8)
tff(fact_9845_and__not__num_Osimps_I4_J,axiom,
! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).
% and_not_num.simps(4)
tff(fact_9846_and__not__num_Osimps_I2_J,axiom,
! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).
% and_not_num.simps(2)
tff(fact_9847_and__not__num_Osimps_I3_J,axiom,
! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).
% and_not_num.simps(3)
tff(fact_9848_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_sx(Mb),Nb) ).
% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9849_and__not__num_Osimps_I7_J,axiom,
! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).
% and_not_num.simps(7)
tff(fact_9850_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_sy(Mb),Nb) ).
% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9851_and__not__num__eq__None__iff,axiom,
! [Mb: num,Nb: num] :
( ( bit_and_not_num(Mb,Nb) = none_num )
<=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).
% and_not_num_eq_None_iff
tff(fact_9852_Bit__Operations_Otake__bit__num__code,axiom,
! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_tc,product_Pair_nat_num(Nb,Mb)) ).
% Bit_Operations.take_bit_num_code
tff(fact_9853_Rat_Opositive_Orep__eq,axiom,
! [Xb: rat] :
( aa_rat_o(positive,Xb)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xb))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xb)))) ) ).
% Rat.positive.rep_eq
tff(fact_9854_and__not__num_Oelims,axiom,
! [Xb: num,Xaa: num,Y: option_num] :
( ( bit_and_not_num(Xb,Xaa) = Y )
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( Y != none_num ) ) )
=> ( ( ( Xb = one )
=> ( ? [N: num] : Xaa = aa_num_num(bit0,N)
=> ( Y != some_num(one) ) ) )
=> ( ( ( Xb = one )
=> ( ? [N: num] : Xaa = aa_num_num(bit1,N)
=> ( Y != none_num ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ( ( Xaa = one )
=> ( Y != some_num(aa_num_num(bit0,M2)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ( ( Xaa = one )
=> ( Y != some_num(aa_num_num(bit0,M2)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_sw,bit_and_not_num(M2,N)) ) ) )
=> ~ ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
tff(fact_9855_xor__num_Osimps_I6_J,axiom,
! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(Mb,Nb))) ).
% xor_num.simps(6)
tff(fact_9856_xor__num_Osimps_I9_J,axiom,
! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_un2480387367778600638or_num(Mb,Nb)) ).
% xor_num.simps(9)
tff(fact_9857_xor__num_Osimps_I5_J,axiom,
! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_un2480387367778600638or_num(Mb,Nb)) ).
% xor_num.simps(5)
tff(fact_9858_and__not__num_Osimps_I5_J,axiom,
! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).
% and_not_num.simps(5)
tff(fact_9859_and__not__num_Osimps_I6_J,axiom,
! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).
% and_not_num.simps(6)
tff(fact_9860_and__not__num_Osimps_I9_J,axiom,
! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).
% and_not_num.simps(9)
tff(fact_9861_xor__num_Osimps_I1_J,axiom,
bit_un2480387367778600638or_num(one,one) = none_num ).
% xor_num.simps(1)
tff(fact_9862_xor__num_Oelims,axiom,
! [Xb: num,Xaa: num,Y: option_num] :
( ( bit_un2480387367778600638or_num(Xb,Xaa) = Y )
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( Y != none_num ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != some_num(aa_num_num(bit1,N)) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != some_num(aa_num_num(bit0,N)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ( ( Xaa = one )
=> ( Y != some_num(aa_num_num(bit1,M2)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != map_option_num_num(bit0,bit_un2480387367778600638or_num(M2,N)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M2,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ( ( Xaa = one )
=> ( Y != some_num(aa_num_num(bit0,M2)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M2,N))) ) ) )
=> ~ ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != map_option_num_num(bit0,bit_un2480387367778600638or_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
tff(fact_9863_xor__num_Osimps_I7_J,axiom,
! [Mb: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).
% xor_num.simps(7)
tff(fact_9864_xor__num_Osimps_I4_J,axiom,
! [Mb: num] : bit_un2480387367778600638or_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit1,Mb)) ).
% xor_num.simps(4)
tff(fact_9865_xor__num_Osimps_I3_J,axiom,
! [Nb: num] : bit_un2480387367778600638or_num(one,aa_num_num(bit1,Nb)) = some_num(aa_num_num(bit0,Nb)) ).
% xor_num.simps(3)
tff(fact_9866_xor__num_Osimps_I2_J,axiom,
! [Nb: num] : bit_un2480387367778600638or_num(one,aa_num_num(bit0,Nb)) = some_num(aa_num_num(bit1,Nb)) ).
% xor_num.simps(2)
tff(fact_9867_xor__num_Osimps_I8_J,axiom,
! [Mb: num,Nb: num] : bit_un2480387367778600638or_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(Mb,Nb))) ).
% xor_num.simps(8)
tff(fact_9868_and__num_Oelims,axiom,
! [Xb: num,Xaa: num,Y: option_num] :
( ( bit_un7362597486090784418nd_num(Xb,Xaa) = Y )
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( Y != some_num(one) ) ) )
=> ( ( ( Xb = one )
=> ( ? [N: num] : Xaa = aa_num_num(bit0,N)
=> ( Y != none_num ) ) )
=> ( ( ( Xb = one )
=> ( ? [N: num] : Xaa = aa_num_num(bit1,N)
=> ( Y != some_num(one) ) ) )
=> ( ( ? [M2: num] : Xb = aa_num_num(bit0,M2)
=> ( ( Xaa = one )
=> ( Y != none_num ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M2,N)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M2,N)) ) ) )
=> ( ( ? [M2: num] : Xb = aa_num_num(bit1,M2)
=> ( ( Xaa = one )
=> ( Y != some_num(one) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M2,N)) ) ) )
=> ~ ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_sw,bit_un7362597486090784418nd_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
tff(fact_9869_and__num_Osimps_I1_J,axiom,
bit_un7362597486090784418nd_num(one,one) = some_num(one) ).
% and_num.simps(1)
tff(fact_9870_and__num_Osimps_I5_J,axiom,
! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_un7362597486090784418nd_num(Mb,Nb)) ).
% and_num.simps(5)
tff(fact_9871_and__num_Osimps_I7_J,axiom,
! [Mb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,Mb),one) = some_num(one) ).
% and_num.simps(7)
tff(fact_9872_and__num_Osimps_I3_J,axiom,
! [Nb: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit1,Nb)) = some_num(one) ).
% and_num.simps(3)
tff(fact_9873_and__num_Osimps_I2_J,axiom,
! [Nb: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit0,Nb)) = none_num ).
% and_num.simps(2)
tff(fact_9874_and__num_Osimps_I4_J,axiom,
! [Mb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),one) = none_num ).
% and_num.simps(4)
tff(fact_9875_and__num_Osimps_I8_J,axiom,
! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_un7362597486090784418nd_num(Mb,Nb)) ).
% and_num.simps(8)
tff(fact_9876_and__num_Osimps_I6_J,axiom,
! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_un7362597486090784418nd_num(Mb,Nb)) ).
% and_num.simps(6)
tff(fact_9877_and__num_Osimps_I9_J,axiom,
! [Mb: num,Nb: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_sw,bit_un7362597486090784418nd_num(Mb,Nb)) ).
% and_num.simps(9)
tff(fact_9878_min__Suc__Suc,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).
% min_Suc_Suc
tff(fact_9879_min__0R,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).
% min_0R
tff(fact_9880_min__0L,axiom,
! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).
% min_0L
tff(fact_9881_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q3),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).
% min_enat_simps(2)
tff(fact_9882_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q3) = zero_z5237406670263579293d_enat ).
% min_enat_simps(3)
tff(fact_9883_min__Suc__numeral,axiom,
! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K))) ).
% min_Suc_numeral
tff(fact_9884_min__numeral__Suc,axiom,
! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Nb)) ).
% min_numeral_Suc
tff(fact_9885_min__diff,axiom,
! [Mb: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),I) ).
% min_diff
tff(fact_9886_nat__mult__min__left,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).
% nat_mult_min_left
tff(fact_9887_nat__mult__min__right,axiom,
! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).
% nat_mult_min_right
tff(fact_9888_min__Suc2,axiom,
! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_td,Nb)),Mb) ).
% min_Suc2
tff(fact_9889_min__Suc1,axiom,
! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_te,Nb)),Mb) ).
% min_Suc1
tff(fact_9890_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
tff(fact_9891_inf__enat__def,axiom,
inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% inf_enat_def
tff(fact_9892_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_9893_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_9894_upto__aux__rec,axiom,
! [I: int,J: int,Js: list_int] :
upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).
% upto_aux_rec
tff(fact_9895_range__mod,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tf,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).
% range_mod
tff(fact_9896_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_9897_upto_Opsimps,axiom,
! [I: int,J: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J))
=> ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).
% upto.psimps
tff(fact_9898_upto_Opelims,axiom,
! [Xb: int,Xaa: int,Y: list_int] :
( ( upto(Xb,Xaa) = Y )
=> ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
=> ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
=> ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).
% upto.pelims
tff(fact_9899_card__UNIV__unit,axiom,
finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).
% card_UNIV_unit
tff(fact_9900_range__mult,axiom,
! [A: real] :
image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).
% range_mult
tff(fact_9901_card__UNIV__bool,axiom,
finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% card_UNIV_bool
tff(fact_9902_upto__Nil,axiom,
! [I: int,J: int] :
( ( upto(I,J) = nil_int )
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).
% upto_Nil
tff(fact_9903_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int = upto(I,J) )
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).
% upto_Nil2
tff(fact_9904_upto__empty,axiom,
! [J: int,I: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
=> ( upto(I,J) = nil_int ) ) ).
% upto_empty
tff(fact_9905_upto__single,axiom,
! [I: int] : upto(I,I) = cons_int(I,nil_int) ).
% upto_single
tff(fact_9906_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
=> ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).
% nth_upto
tff(fact_9907_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_9908_upto__rec__numeral_I1_J,axiom,
! [Mb: num,Nb: num] :
upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).
% upto_rec_numeral(1)
tff(fact_9909_upto__rec__numeral_I4_J,axiom,
! [Mb: num,Nb: num] :
upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).
% upto_rec_numeral(4)
tff(fact_9910_upto__rec__numeral_I3_J,axiom,
! [Mb: num,Nb: num] :
upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).
% upto_rec_numeral(3)
tff(fact_9911_upto__rec__numeral_I2_J,axiom,
! [Mb: num,Nb: num] :
upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).
% upto_rec_numeral(2)
tff(fact_9912_upto__aux__def,axiom,
! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).
% upto_aux_def
tff(fact_9913_upto__code,axiom,
! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).
% upto_code
tff(fact_9914_distinct__upto,axiom,
! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).
% distinct_upto
tff(fact_9915_atLeastAtMost__upto,axiom,
! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).
% atLeastAtMost_upto
tff(fact_9916_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_9917_upto__split2,axiom,
! [I: int,J: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
=> ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).
% upto_split2
tff(fact_9918_upto__split1,axiom,
! [I: int,J: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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_9919_upto__rec1,axiom,
! [I: int,J: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
=> ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).
% upto_rec1
tff(fact_9920_upto_Oelims,axiom,
! [Xb: int,Xaa: int,Y: list_int] :
( ( upto(Xb,Xaa) = Y )
=> ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).
% upto.elims
tff(fact_9921_upto_Osimps,axiom,
! [I: int,J: int] :
upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).
% upto.simps
tff(fact_9922_upto__rec2,axiom,
! [I: int,J: int] :
( aa_int_o(aa_int_fun_int_o(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_9923_upto__split3,axiom,
! [I: int,J: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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_9924_root__def,axiom,
! [Nb: nat,Xb: real] :
aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_tg,Nb)),Xb)) ).
% root_def
tff(fact_9925_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_9926_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_9927_char_Osize_I2_J,axiom,
! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).
% char.size(2)
tff(fact_9928_nat__of__char__less__256,axiom,
! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),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_9929_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_9930_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : 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_9931_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_9932_DERIV__even__real__root,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),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,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).
% DERIV_even_real_root
tff(fact_9933_DERIV__pos__inc__right,axiom,
! [F: fun_real_real,L: real,Xb: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ).
% DERIV_pos_inc_right
tff(fact_9934_DERIV__neg__dec__right,axiom,
! [F: fun_real_real,L: real,Xb: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).
% DERIV_neg_dec_right
tff(fact_9935_DERIV__const__ratio__const,axiom,
! [A: real,B: real,F: fun_real_real,K: real] :
( ( A != B )
=> ( ! [X4: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X4,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_9936_DERIV__neg__dec__left,axiom,
! [F: fun_real_real,L: real,Xb: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ).
% DERIV_neg_dec_left
tff(fact_9937_DERIV__pos__inc__left,axiom,
! [F: fun_real_real,L: real,Xb: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).
% DERIV_pos_inc_left
tff(fact_9938_DERIV__neg__imp__decreasing,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
=> ? [Y4: real] :
( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).
% DERIV_neg_imp_decreasing
tff(fact_9939_DERIV__pos__imp__increasing,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
=> ? [Y4: real] :
( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).
% DERIV_pos_imp_increasing
tff(fact_9940_has__real__derivative__pos__inc__left,axiom,
! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3)),S3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
tff(fact_9941_has__real__derivative__neg__dec__left,axiom,
! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3)),S3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
tff(fact_9942_has__real__derivative__pos__inc__right,axiom,
! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3)),S3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
tff(fact_9943_has__real__derivative__neg__dec__right,axiom,
! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
=> ? [D4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
& ! [H3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
=> ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3)),S3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
tff(fact_9944_MVT2,axiom,
! [A: real,B: real,F: fun_real_real,F3: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
=> has_fi5821293074295781190e_real(F,aa_real_real(F3,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ? [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),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,Z2)) ) ) ) ) ).
% MVT2
tff(fact_9945_DERIV__local__const,axiom,
! [F: fun_real_real,L: real,Xb: real,D: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
=> ( ! [Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
=> ( aa_real_real(F,Xb) = aa_real_real(F,Y3) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_const
tff(fact_9946_DERIV__ln,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).
% DERIV_ln
tff(fact_9947_DERIV__const__average,axiom,
! [A: real,B: real,V: fun_real_real,K: real] :
( ( A != B )
=> ( ! [X4: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X4,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)),aa_num_real(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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).
% DERIV_const_average
tff(fact_9948_DERIV__local__min,axiom,
! [F: fun_real_real,L: real,Xb: real,D: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
=> ( ! [Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xb)),aa_real_real(F,Y3)) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_min
tff(fact_9949_DERIV__local__max,axiom,
! [F: fun_real_real,L: real,Xb: real,D: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
=> ( ! [Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,Xb)) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_max
tff(fact_9950_DERIV__ln__divide,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).
% DERIV_ln_divide
tff(fact_9951_DERIV__pow,axiom,
! [Nb: nat,Xb: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_th,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S2)) ).
% DERIV_pow
tff(fact_9952_DERIV__fun__pow,axiom,
! [G: fun_real_real,Mb: real,Xb: real,Nb: nat] :
( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ti(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).
% DERIV_fun_pow
tff(fact_9953_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
=> has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_tj,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).
% has_real_derivative_powr
tff(fact_9954_DERIV__fun__powr,axiom,
! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
=> has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_tk(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).
% DERIV_fun_powr
tff(fact_9955_DERIV__log,axiom,
! [Xb: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> 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)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).
% DERIV_log
tff(fact_9956_DERIV__powr,axiom,
! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
=> ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
=> has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_tl(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(F,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).
% DERIV_powr
tff(fact_9957_DERIV__real__sqrt,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).
% DERIV_real_sqrt
tff(fact_9958_DERIV__arctan,axiom,
! [Xb: 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).
% DERIV_arctan
tff(fact_9959_arsinh__real__has__field__derivative,axiom,
! [Xb: real,A3: 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ).
% arsinh_real_has_field_derivative
tff(fact_9960_DERIV__real__sqrt__generic,axiom,
! [Xb: real,D5: real] :
( ( Xb != zero_zero_real )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> ( D5 = 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,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
=> has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).
% DERIV_real_sqrt_generic
tff(fact_9961_arcosh__real__has__field__derivative,axiom,
! [Xb: real,A3: set_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ) ).
% arcosh_real_has_field_derivative
tff(fact_9962_artanh__real__has__field__derivative,axiom,
! [Xb: real,A3: set_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,A3)) ) ).
% artanh_real_has_field_derivative
tff(fact_9963_DERIV__real__root,axiom,
! [Nb: nat,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).
% DERIV_real_root
tff(fact_9964_DERIV__arccos,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).
% DERIV_arccos
tff(fact_9965_DERIV__arcsin,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).
% DERIV_arcsin
tff(fact_9966_Maclaurin__all__le__objl,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
& ! [M2: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tm(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).
% Maclaurin_all_le_objl
tff(fact_9967_Maclaurin__all__le,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tm(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).
% Maclaurin_all_le
tff(fact_9968_DERIV__odd__real__root,axiom,
! [Nb: nat,Xb: real] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( ( Xb != zero_zero_real )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).
% DERIV_odd_real_root
tff(fact_9969_Maclaurin,axiom,
! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),H)
& ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).
% Maclaurin
tff(fact_9970_Maclaurin2,axiom,
! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H)
& ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).
% Maclaurin2
tff(fact_9971_Maclaurin__minus,axiom,
! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),zero_zero_real) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),zero_zero_real)
& ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).
% Maclaurin_minus
tff(fact_9972_Maclaurin__all__lt,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( Xb != zero_zero_real )
=> ( ! [M2: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2))
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tm(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).
% Maclaurin_all_lt
tff(fact_9973_Maclaurin__bi__le,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb)) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
& ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tm(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).
% Maclaurin_bi_le
tff(fact_9974_Taylor,axiom,
! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xb: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),B) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
=> ( ( Xb != C )
=> ? [T2: real] :
( $ite(
aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),C) ),
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Xb) ) )
& ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_tn(Diff),C),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
tff(fact_9975_Taylor__up,axiom,
! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),B) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),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_to(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Nb))) ) ) ) ) ) ) ) ).
% Taylor_up
tff(fact_9976_Taylor__down,axiom,
! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),B) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
=> ? [T2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,T2),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_to(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Nb))) ) ) ) ) ) ) ) ).
% Taylor_down
tff(fact_9977_Maclaurin__lemma2,axiom,
! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B3: real] :
( ! [M2: nat,T2: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
=> ( ( Nb = aa_nat_nat(suc,K) )
=> ! [M3: nat,T5: real] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_tq(Nb),Diff),B3),M3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tr(Diff),M3),T5),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M3))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M3)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M3))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).
% Maclaurin_lemma2
tff(fact_9978_DERIV__arctan__series,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> has_fi5821293074295781190e_real(aTP_Lamm_ts,suminf_real(aa_real_fun_nat_real(aTP_Lamm_tt,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).
% DERIV_arctan_series
tff(fact_9979_DERIV__real__root__generic,axiom,
! [Nb: nat,Xb: real,D5: real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( ( Xb != zero_zero_real )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
=> ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
=> ( D5 = 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,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D5,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).
% DERIV_real_root_generic
tff(fact_9980_DERIV__power__series_H,axiom,
! [R: real,F: fun_nat_real,X0: real] :
( ! [X4: real] :
( aa_set_real_o(member_real(X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_tu(F),X4)) )
=> ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
=> has_fi5821293074295781190e_real(aTP_Lamm_tw(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_tu(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).
% DERIV_power_series'
tff(fact_9981_DERIV__isconst3,axiom,
! [A: real,B: real,Xb: real,Y: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( aa_set_real_o(member_real(Xb),set_or1633881224788618240n_real(A,B))
=> ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,B))
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),set_or1633881224788618240n_real(A,B))
=> has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) )
=> ( aa_real_real(F,Xb) = aa_real_real(F,Y) ) ) ) ) ) ).
% DERIV_isconst3
tff(fact_9982_DERIV__series_H,axiom,
! [F: fun_re4238438383274954274t_real,F3: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_tx(F),N),aa_nat_real(aa_real_fun_nat_real(F3,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),set_or1633881224788618240n_real(A,B))
=> summable_real(aa_real_fun_nat_real(F,X4)) )
=> ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A,B))
=> ( summable_real(aa_real_fun_nat_real(F3,X0))
=> ( summable_real(L5)
=> ( ! [N: nat,X4: real,Y3: real] :
( aa_set_real_o(member_real(X4),set_or1633881224788618240n_real(A,B))
=> ( aa_set_real_o(member_real(Y3),set_or1633881224788618240n_real(A,B))
=> aa_real_o(aa_real_fun_real_o(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,X4),N)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),Y3)))) ) )
=> has_fi5821293074295781190e_real(aTP_Lamm_ty(F),suminf_real(aa_real_fun_nat_real(F3,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).
% DERIV_series'
tff(fact_9983_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_9984_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_9985_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))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
=> ? [R3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
& ! [X3: real] :
( ( ( X3 != C )
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R3) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),zero_zero_real) ) ) ) ) ).
% LIM_fun_less_zero
tff(fact_9986_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] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
& ! [X3: real] :
( ( ( X3 != C )
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R3) )
=> ( aa_real_real(F,X3) != zero_zero_real ) ) ) ) ) ).
% LIM_fun_not_zero
tff(fact_9987_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))
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
=> ? [R3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
& ! [X3: real] :
( ( ( X3 != C )
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R3) )
=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X3)) ) ) ) ) ).
% LIM_fun_gt_zero
tff(fact_9988_isCont__inverse__function2,axiom,
! [A: real,Xb: real,B: real,G: fun_real_real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
=> ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ) ).
% isCont_inverse_function2
tff(fact_9989_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
=> ( linord2614967742042102400et_nat(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_9990_isCont__arcosh,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcosh_real) ) ).
% isCont_arcosh
tff(fact_9991_LIM__cos__div__sin,axiom,
filterlim_real_real(aTP_Lamm_tz,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).
% LIM_cos_div_sin
tff(fact_9992_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [Nb: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9993_DERIV__inverse__function,axiom,
! [F: fun_real_real,D5: real,G: fun_real_real,Xb: real,A: real,B: real] :
( has_fi5821293074295781190e_real(F,D5,topolo2177554685111907308n_real(aa_real_real(G,Xb),top_top_set_real))
=> ( ( D5 != zero_zero_real )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
=> ( ! [Y3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y3)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B)
=> ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
=> ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),G)
=> has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D5),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ).
% DERIV_inverse_function
tff(fact_9994_isCont__arccos,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arccos) ) ) ).
% isCont_arccos
tff(fact_9995_isCont__arcsin,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcsin) ) ) ).
% isCont_arcsin
tff(fact_9996_LIM__less__bound,axiom,
! [B: real,Xb: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xb)
=> ( ! [X4: real] :
( aa_set_real_o(member_real(X4),set_or1633881224788618240n_real(B,Xb))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
=> ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),F)
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xb)) ) ) ) ).
% LIM_less_bound
tff(fact_9997_isCont__artanh,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),artanh_real) ) ) ).
% isCont_artanh
tff(fact_9998_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_9999_isCont__inverse__function,axiom,
! [D: real,Xb: real,G: fun_real_real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),Xb))),D)
=> ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),Xb))),D)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ).
% isCont_inverse_function
tff(fact_10000_GMVT_H,axiom,
! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
=> has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
=> ( ! [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
=> has_fi5821293074295781190e_real(F,aa_real_real(F3,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
=> ? [C3: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
& aa_real_o(aa_real_fun_real_o(ord_less_real,C3),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,C3)) = 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,C3)) ) ) ) ) ) ) ) ).
% GMVT'
tff(fact_10001_summable__Leibniz_I3_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
=> ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ua(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ua(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))),N5)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_ua(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))),N5))))) ) ) ) ).
% summable_Leibniz(3)
tff(fact_10002_summable__Leibniz_I2_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
=> ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ua(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ua(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))),N5))),groups6591440286371151544t_real(aTP_Lamm_ua(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))),N5)),one_one_nat))))) ) ) ) ).
% summable_Leibniz(2)
tff(fact_10003_mult__nat__right__at__top,axiom,
! [C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_au,C),at_top_nat,at_top_nat) ) ).
% mult_nat_right_at_top
tff(fact_10004_mult__nat__left__at__top,axiom,
! [C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).
% mult_nat_left_at_top
tff(fact_10005_nested__sequence__unique,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
=> ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ub(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ? [L3: real] :
( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N5)),L3)
& filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
& ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N5))
& filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).
% nested_sequence_unique
tff(fact_10006_LIMSEQ__inverse__zero,axiom,
! [X6: fun_nat_real] :
( ! [R3: real] :
? [N6: nat] :
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X6,N)) )
=> filterlim_nat_real(aTP_Lamm_uc(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_inverse_zero
tff(fact_10007_LIMSEQ__root__const,axiom,
! [C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ud,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).
% LIMSEQ_root_const
tff(fact_10008_LIMSEQ__inverse__real__of__nat,axiom,
filterlim_nat_real(aTP_Lamm_ue,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat
tff(fact_10009_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uf,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat_add
tff(fact_10010_increasing__LIMSEQ,axiom,
! [F: fun_nat_real,L: real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
=> ( ! [E2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
=> ? [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E2)) )
=> filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).
% increasing_LIMSEQ
tff(fact_10011_LIMSEQ__realpow__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
=> filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).
% LIMSEQ_realpow_zero
tff(fact_10012_LIMSEQ__divide__realpow__zero,axiom,
! [Xb: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ug(Xb),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_divide_realpow_zero
tff(fact_10013_LIMSEQ__abs__realpow__zero,axiom,
! [C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
=> filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_abs_realpow_zero
tff(fact_10014_LIMSEQ__abs__realpow__zero2,axiom,
! [C: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
=> filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_abs_realpow_zero2
tff(fact_10015_LIMSEQ__inverse__realpow__zero,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uh,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_inverse_realpow_zero
tff(fact_10016_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ui,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10017_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uj,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10018_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_ua(A)) ) ) ).
% summable_Leibniz(1)
tff(fact_10019_summable,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
=> summable_real(aTP_Lamm_ua(A)) ) ) ) ).
% summable
tff(fact_10020_cos__diff__limit__1,axiom,
! [Theta: fun_nat_real,Theta2: real] :
( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uk(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
=> ~ ! [K3: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_ul(Theta),K3),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).
% cos_diff_limit_1
tff(fact_10021_cos__limit__1,axiom,
! [Theta: fun_nat_real] :
( filterlim_nat_real(aTP_Lamm_um(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
=> ? [K3: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_ul(Theta),K3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% cos_limit_1
tff(fact_10022_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_un(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ua(A))),at_top_nat) ) ) ).
% summable_Leibniz(4)
tff(fact_10023_zeroseq__arctan__series,axiom,
! [Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_id,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% zeroseq_arctan_series
tff(fact_10024_summable__Leibniz_H_I3_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
=> filterlim_nat_real(aTP_Lamm_un(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ua(A))),at_top_nat) ) ) ) ).
% summable_Leibniz'(3)
tff(fact_10025_summable__Leibniz_H_I2_J,axiom,
! [A: fun_nat_real,Nb: nat] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ua(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))),Nb)))),suminf_real(aTP_Lamm_ua(A))) ) ) ) ).
% summable_Leibniz'(2)
tff(fact_10026_sums__alternating__upper__lower,axiom,
! [A: fun_nat_real] :
( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
=> ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ? [L3: real] :
( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ua(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))),N5)))),L3)
& filterlim_nat_real(aTP_Lamm_un(A),topolo2815343760600316023s_real(L3),at_top_nat)
& ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_ua(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))),N5)),one_one_nat))))
& filterlim_nat_real(aTP_Lamm_uo(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).
% sums_alternating_upper_lower
tff(fact_10027_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_uo(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ua(A))),at_top_nat) ) ) ).
% summable_Leibniz(5)
tff(fact_10028_summable__Leibniz_H_I5_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
=> filterlim_nat_real(aTP_Lamm_uo(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ua(A))),at_top_nat) ) ) ) ).
% summable_Leibniz'(5)
tff(fact_10029_summable__Leibniz_H_I4_J,axiom,
! [A: fun_nat_real,Nb: nat] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
=> ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_ua(A))),groups6591440286371151544t_real(aTP_Lamm_ua(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))),Nb)),one_one_nat)))) ) ) ) ).
% summable_Leibniz'(4)
tff(fact_10030_real__bounded__linear,axiom,
! [F: fun_real_real] :
( real_V5970128139526366754l_real(F)
<=> ? [C2: real] :
! [X: real] : aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),C2) ) ).
% real_bounded_linear
tff(fact_10031_filterlim__Suc,axiom,
filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).
% filterlim_Suc
tff(fact_10032_tendsto__exp__limit__at__right,axiom,
! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_up,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).
% tendsto_exp_limit_at_right
tff(fact_10033_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),aa_num_real(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).
% filterlim_tan_at_right
tff(fact_10034_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).
% tendsto_arctan_at_bot
tff(fact_10035_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_10036_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_10037_DERIV__pos__imp__increasing__at__bot,axiom,
! [B: real,F: fun_real_real,Flim: real] :
( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
=> ? [Y4: real] :
( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) )
=> ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).
% DERIV_pos_imp_increasing_at_bot
tff(fact_10038_filterlim__pow__at__bot__odd,axiom,
! [Nb: nat,F: fun_real_real,F4: filter_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( filterlim_real_real(F,at_bot_real,F4)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_uq(Nb),F),at_bot_real,F4) ) ) ) ).
% filterlim_pow_at_bot_odd
tff(fact_10039_filterlim__pow__at__bot__even,axiom,
! [Nb: nat,F: fun_real_real,F4: filter_real] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> ( filterlim_real_real(F,at_bot_real,F4)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_uq(Nb),F),at_top_real,F4) ) ) ) ).
% filterlim_pow_at_bot_even
tff(fact_10040_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).
% filterlim_tan_at_left
tff(fact_10041_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F: fun_real_real,Flim: real] :
( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X4)
=> ? [Y4: real] :
( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) )
=> ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).
% DERIV_neg_imp_decreasing_at_top
tff(fact_10042_tendsto__arctan__at__top,axiom,
filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).
% tendsto_arctan_at_top
tff(fact_10043_GMVT,axiom,
! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [X4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
=> ( ! [X4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B) )
=> differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
=> ( ! [X4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
=> ( ! [X4: real] :
( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B) )
=> differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,top_top_set_real)) )
=> ? [G_c: real,F_c: real,C3: real] :
( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
& has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
& aa_real_o(aa_real_fun_real_o(ord_less_real,C3),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_10044_eventually__sequentially__Suc,axiom,
! [P: fun_nat_o] :
( eventually_nat(aTP_Lamm_ur(P),at_top_nat)
<=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentially_Suc
tff(fact_10045_eventually__sequentially__seg,axiom,
! [P: fun_nat_o,K: nat] :
( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_us(P),K),at_top_nat)
<=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentially_seg
tff(fact_10046_sequentially__offset,axiom,
! [P: fun_nat_o,K: nat] :
( eventually_nat(P,at_top_nat)
=> eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_us(P),K),at_top_nat) ) ).
% sequentially_offset
tff(fact_10047_eventually__sequentially,axiom,
! [P: fun_nat_o] :
( eventually_nat(P,at_top_nat)
<=> ? [N8: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N3)
=> aa_nat_o(P,N3) ) ) ).
% eventually_sequentially
tff(fact_10048_eventually__sequentiallyI,axiom,
! [C: nat,P: fun_nat_o] :
( ! [X4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
=> aa_nat_o(P,X4) )
=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentiallyI
tff(fact_10049_le__sequentially,axiom,
! [F4: filter_nat] :
( ord_le2510731241096832064er_nat(F4,at_top_nat)
<=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F4) ) ).
% le_sequentially
tff(fact_10050_eventually__at__left__real,axiom,
! [B: real,A: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
=> eventually_real(aa_real_fun_real_o(aTP_Lamm_ut(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).
% eventually_at_left_real
tff(fact_10051_eventually__at__right__real,axiom,
! [A: real,B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> eventually_real(aa_real_fun_real_o(aTP_Lamm_ut(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).
% eventually_at_right_real
tff(fact_10052_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_10053_GreatestI__ex__nat,axiom,
! [P: fun_nat_o,B: nat] :
( ? [X_1: nat] : aa_nat_o(P,X_1)
=> ( ! [Y3: nat] :
( aa_nat_o(P,Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
=> aa_nat_o(P,order_Greatest_nat(P)) ) ) ).
% GreatestI_ex_nat
tff(fact_10054_Greatest__le__nat,axiom,
! [P: fun_nat_o,K: nat,B: nat] :
( aa_nat_o(P,K)
=> ( ! [Y3: nat] :
( aa_nat_o(P,Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).
% Greatest_le_nat
tff(fact_10055_GreatestI__nat,axiom,
! [P: fun_nat_o,K: nat,B: nat] :
( aa_nat_o(P,K)
=> ( ! [Y3: nat] :
( aa_nat_o(P,Y3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
=> aa_nat_o(P,order_Greatest_nat(P)) ) ) ).
% GreatestI_nat
tff(fact_10056_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
=> ( linord2614967742042102400et_nat(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_10057_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [Nb: nat,J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_10058_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_10059_greaterThanAtMost__upto,axiom,
! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).
% greaterThanAtMost_upto
tff(fact_10060_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_10061_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
<=> (Y) )
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> ( (Y)
<=> ( Xaa != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
=> ( (Y)
<=> ~ ( ( Deg = Xaa )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summary),m2)
& ( 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))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
tff(fact_10062_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> ( Xaa != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
=> ~ ( ( Deg = Xaa )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summary),m2)
& ( 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))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
tff(fact_10063_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
<=> ( ( Dega = Deg3 )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeLista))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
& ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeLista))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).
% VEBT_internal.valid'.simps(2)
tff(fact_10064_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
=> ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
=> ( Xaa = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
=> ( ( Deg = Xaa )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summary),m2)
& ( 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))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
tff(fact_10065_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
=> ( Xaa = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
=> ( ( Deg = Xaa )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summary),m2)
& ( 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))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
tff(fact_10066_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat] :
( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
=> ( Xaa != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
=> ~ ( ( Deg = Xaa )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summary),m2)
& ( 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))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
tff(fact_10067_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
<=> (Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
=> ( ! [Uu: $o,Uv: $o] :
( ( Xb = vEBT_Leaf((Uu),(Uv)) )
=> ( ( (Y)
<=> ( Xaa = one_one_nat ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
=> ( ( (Y)
<=> ( ( Deg = Xaa )
& $let(
n: nat,
n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
$let(
m2: nat,
m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
( ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> aa_nat_o(vEBT_VEBT_valid(X),n) )
& aa_nat_o(vEBT_VEBT_valid(Summary),m2)
& ( 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))),m2) )
& aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
& ! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
tff(fact_10068_MVT,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ? [L3: real,Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
& has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z2,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_10069_Rolle__deriv,axiom,
! [A: real,B: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_real(F,A) = aa_real_real(F,B) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ? [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
& ! [X3: real] : aa_real_real(aa_rea7340605660300375871l_real(F3,Z2),X3) = zero_zero_real ) ) ) ) ) ).
% Rolle_deriv
tff(fact_10070_mvt,axiom,
! [A: real,B: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ~ ! [Xi: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
=> ( aa_real_o(aa_real_fun_real_o(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_10071_DERIV__pos__imp__increasing__open,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> ? [Y4: real] :
( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ) ).
% DERIV_pos_imp_increasing_open
tff(fact_10072_DERIV__neg__imp__decreasing__open,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> ? [Y4: real] :
( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
& aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ) ).
% DERIV_neg_imp_decreasing_open
tff(fact_10073_DERIV__isconst__end,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).
% DERIV_isconst_end
tff(fact_10074_DERIV__isconst2,axiom,
! [A: real,B: real,F: fun_real_real,Xb: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
=> ( aa_real_real(F,Xb) = aa_real_real(F,A) ) ) ) ) ) ) ).
% DERIV_isconst2
tff(fact_10075_Rolle,axiom,
! [A: real,B: real,F: fun_real_real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
=> ( ( aa_real_real(F,A) = aa_real_real(F,B) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X4: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
=> differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
=> ? [Z2: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
& has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).
% Rolle
tff(fact_10076_mono__Suc,axiom,
order_mono_nat_nat(suc) ).
% mono_Suc
tff(fact_10077_mono__times__nat,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).
% mono_times_nat
tff(fact_10078_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_uv,K)) ) ).
% mono_ge2_power_minus_self
tff(fact_10079_inj__sgn__power,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_tg,Nb)),top_top_set_real) ) ).
% inj_sgn_power
tff(fact_10080_log__inj,axiom,
! [B: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
=> aa_set_real_o(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real)) ) ).
% log_inj
tff(fact_10081_inj__Suc,axiom,
! [N4: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N4) ).
% inj_Suc
tff(fact_10082_inj__on__diff__nat,axiom,
! [N4: set_nat,K: nat] :
( ! [N: nat] :
( aa_set_nat_o(member_nat(N),N4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
=> aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,K)),N4) ) ).
% inj_on_diff_nat
tff(fact_10083_inj__on__char__of__nat,axiom,
aa_set_nat_o(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_10084_sup__enat__def,axiom,
sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% sup_enat_def
tff(fact_10085_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
tff(fact_10086_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_10087_remdups__upt,axiom,
! [Mb: nat,Nb: nat] : remdups_nat(upt(Mb,Nb)) = upt(Mb,Nb) ).
% remdups_upt
tff(fact_10088_hd__upt,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( hd_nat(upt(I,J)) = I ) ) ).
% hd_upt
tff(fact_10089_drop__upt,axiom,
! [Mb: nat,I: nat,J: nat] : drop_nat(Mb,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb),J) ).
% drop_upt
tff(fact_10090_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_10091_take__upt,axiom,
! [I: nat,Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)),Nb)
=> ( take_nat(Mb,upt(I,Nb)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)) ) ) ).
% take_upt
tff(fact_10092_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
=> ( upt(I,J) = nil_nat ) ) ).
% upt_conv_Nil
tff(fact_10093_sorted__list__of__set__range,axiom,
! [Mb: nat,Nb: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = upt(Mb,Nb) ).
% sorted_list_of_set_range
tff(fact_10094_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( upt(I,J) = nil_nat )
<=> ( ( J = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).
% upt_eq_Nil_conv
tff(fact_10095_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
=> ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).
% nth_upt
tff(fact_10096_upt__rec__numeral,axiom,
! [Mb: num,Nb: num] :
upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).
% upt_rec_numeral
tff(fact_10097_map__Suc__upt,axiom,
! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).
% map_Suc_upt
tff(fact_10098_map__add__upt,axiom,
! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).
% map_add_upt
tff(fact_10099_distinct__upt,axiom,
! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).
% distinct_upt
tff(fact_10100_greaterThanLessThan__upt,axiom,
! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).
% greaterThanLessThan_upt
tff(fact_10101_atLeastLessThan__upt,axiom,
! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).
% atLeastLessThan_upt
tff(fact_10102_atLeastAtMost__upt,axiom,
! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).
% atLeastAtMost_upt
tff(fact_10103_atLeast__upt,axiom,
! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).
% atLeast_upt
tff(fact_10104_greaterThanAtMost__upt,axiom,
! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).
% greaterThanAtMost_upt
tff(fact_10105_upt__conv__Cons__Cons,axiom,
! [Mb: nat,Nb: nat,Ns: list_nat,Q3: nat] :
( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q3) )
<=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q3) ) ) ).
% upt_conv_Cons_Cons
tff(fact_10106_upt__0,axiom,
! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).
% upt_0
tff(fact_10107_atMost__upto,axiom,
! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).
% atMost_upto
tff(fact_10108_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).
% upt_conv_Cons
tff(fact_10109_map__decr__upt,axiom,
! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_ux,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).
% map_decr_upt
tff(fact_10110_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_10111_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,Xb: nat,Xs: list_nat] :
( ( upt(I,J) = cons_nat(Xb,Xs) )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& ( I = Xb )
& ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_10112_upt__rec,axiom,
! [I: nat,J: nat] :
upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).
% upt_rec
tff(fact_10113_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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_10114_upt__Suc,axiom,
! [I: nat,J: nat] :
upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).
% upt_Suc
tff(fact_10115_tl__upt,axiom,
! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).
% tl_upt
tff(fact_10116_sum__list__upt,axiom,
! [Mb: nat,Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
=> ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aj),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) ) ) ).
% sum_list_upt
tff(fact_10117_card__length__sum__list__rec,axiom,
! [Mb: nat,N4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
=> ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uy(Mb),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uz(Mb),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_va(Mb),N4)))) ) ) ).
% card_length_sum_list_rec
tff(fact_10118_card__length__sum__list,axiom,
! [Mb: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uy(Mb),N4))) = 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,N4),Mb)),one_one_nat)),N4) ).
% card_length_sum_list
tff(fact_10119_sorted__upt,axiom,
! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).
% sorted_upt
tff(fact_10120_sorted__wrt__upt,axiom,
! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Nb)) ).
% sorted_wrt_upt
tff(fact_10121_sorted__wrt__upto,axiom,
! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).
% sorted_wrt_upto
tff(fact_10122_sorted__upto,axiom,
! [Mb: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Mb,Nb)) ).
% sorted_upto
tff(fact_10123_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).
% sorted_wrt_less_idx
tff(fact_10124_powr__real__of__int_H,axiom,
! [Xb: real,Nb: int] :
( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
=> ( ( ( Xb != zero_zero_real )
| aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = aa_int_real(power_int_real(Xb),Nb) ) ) ) ).
% powr_real_of_int'
tff(fact_10125_pred__nat__def,axiom,
pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vb)) ).
% pred_nat_def
tff(fact_10126_less__eq,axiom,
! [Mb: nat,Nb: nat] :
( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi6264000038957366511cl_nat(pred_nat))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).
% less_eq
tff(fact_10127_pred__nat__trancl__eq__le,axiom,
! [Mb: nat,Nb: nat] :
( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi2905341329935302413cl_nat(pred_nat))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).
% pred_nat_trancl_eq_le
tff(fact_10128_pos__deriv__imp__strict__mono,axiom,
! [F: fun_real_real,F3: fun_real_real] :
( ! [X4: real] : has_fi5821293074295781190e_real(F,aa_real_real(F3,X4),topolo2177554685111907308n_real(X4,top_top_set_real))
=> ( ! [X4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F3,X4))
=> order_7092887310737990675l_real(F) ) ) ).
% pos_deriv_imp_strict_mono
tff(fact_10129_pairs__le__eq__Sigma,axiom,
! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_qm(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aTP_Lamm_vc(Mb)) ).
% pairs_le_eq_Sigma
tff(fact_10130_strict__mono__imp__increasing,axiom,
! [F: fun_nat_nat,Nb: nat] :
( order_5726023648592871131at_nat(F)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(F,Nb)) ) ).
% strict_mono_imp_increasing
tff(fact_10131_vimage__Suc__insert__Suc,axiom,
! [Nb: nat,A3: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Nb),A3)) = insert_nat(Nb,vimage_nat_nat(suc,A3)) ).
% vimage_Suc_insert_Suc
tff(fact_10132_finite__vimage__Suc__iff,axiom,
! [F4: set_nat] :
( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F4))
<=> aa_set_nat_o(finite_finite_nat,F4) ) ).
% finite_vimage_Suc_iff
tff(fact_10133_vimage__Suc__insert__0,axiom,
! [A3: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A3)) = vimage_nat_nat(suc,A3) ).
% vimage_Suc_insert_0
tff(fact_10134_set__decode__div__2,axiom,
! [Xb: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,Xb)) ).
% set_decode_div_2
tff(fact_10135_set__encode__vimage__Suc,axiom,
! [A3: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% set_encode_vimage_Suc
tff(fact_10136_Field__natLeq__on,axiom,
! [Nb: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vd(Nb)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Nb)) ).
% Field_natLeq_on
tff(fact_10137_natLess__def,axiom,
bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).
% natLess_def
tff(fact_10138_wf__less,axiom,
wf_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).
% wf_less
tff(fact_10139_Restr__natLeq,axiom,
! [Nb: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Nb)),aTP_Lamm_ve(Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vd(Nb))) ).
% Restr_natLeq
tff(fact_10140_natLeq__def,axiom,
bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).
% natLeq_def
tff(fact_10141_Restr__natLeq2,axiom,
! [Nb: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Nb),aTP_Lamm_vf(Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vd(Nb))) ).
% Restr_natLeq2
tff(fact_10142_prod__encode__prod__decode__aux,axiom,
! [K: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb) ).
% prod_encode_prod_decode_aux
tff(fact_10143_natLeq__underS__less,axiom,
! [Nb: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Nb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Nb)) ).
% natLeq_underS_less
tff(fact_10144_le__prod__encode__1,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(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_10145_le__prod__encode__2,axiom,
! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(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_10146_prod__encode__def,axiom,
nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_vg) ).
% prod_encode_def
tff(fact_10147_list__encode_Oelims,axiom,
! [Xb: list_nat,Y: nat] :
( ( nat_list_encode(Xb) = Y )
=> ( ( ( Xb = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( Xb = cons_nat(X4,Xs2) )
=> ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) ) ) ) ) ).
% list_encode.elims
tff(fact_10148_list__encode_Osimps_I2_J,axiom,
! [Xb: nat,Xs: list_nat] : nat_list_encode(cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),nat_list_encode(Xs)))) ).
% list_encode.simps(2)
tff(fact_10149_list__encode_Opelims,axiom,
! [Xb: list_nat,Y: nat] :
( ( nat_list_encode(Xb) = Y )
=> ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
=> ( ( ( Xb = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( Xb = cons_nat(X4,Xs2) )
=> ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) )
=> ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2)) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_10150_quotient__of__def,axiom,
! [Xb: rat] : quotient_of(Xb) = the_Pr4378521158711661632nt_int(aTP_Lamm_vh(Xb)) ).
% quotient_of_def
tff(fact_10151_normalize__stable,axiom,
! [Q3: int,P2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3)
=> ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q3)
=> ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3) ) ) ) ).
% normalize_stable
tff(fact_10152_coprime__crossproduct__int,axiom,
! [A: int,D: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),D)
=> ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,B),C)
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,B)),aa_int_int(abs_abs_int,D)) )
<=> ( ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) )
& ( aa_int_int(abs_abs_int,C) = aa_int_int(abs_abs_int,D) ) ) ) ) ) ).
% coprime_crossproduct_int
tff(fact_10153_Rat__induct,axiom,
! [P: fun_rat_o,Q3: rat] :
( ! [A4: int,B4: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
=> ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4)
=> aa_rat_o(P,aa_int_rat(fract(A4),B4)) ) )
=> aa_rat_o(P,Q3) ) ).
% Rat_induct
tff(fact_10154_Rat__cases,axiom,
! [Q3: rat] :
~ ! [A4: int,B4: int] :
( ( Q3 = aa_int_rat(fract(A4),B4) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
=> ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) ).
% Rat_cases
tff(fact_10155_Rat__cases__nonzero,axiom,
! [Q3: rat] :
( ! [A4: int,B4: int] :
( ( Q3 = aa_int_rat(fract(A4),B4) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
=> ( ( A4 != zero_zero_int )
=> ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) )
=> ( Q3 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
tff(fact_10156_quotient__of__unique,axiom,
! [R2: rat] :
? [X4: product_prod_int_int] :
( ( R2 = aa_int_rat(fract(product_fst_int_int(X4)),product_snd_int_int(X4)) )
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(X4))
& aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(X4)),product_snd_int_int(X4))
& ! [Y4: product_prod_int_int] :
( ( ( R2 = aa_int_rat(fract(product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Y4))
& aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
=> ( Y4 = X4 ) ) ) ).
% quotient_of_unique
tff(fact_10157_coprime__Suc__0__left,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ).
% coprime_Suc_0_left
tff(fact_10158_coprime__Suc__0__right,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) ).
% coprime_Suc_0_right
tff(fact_10159_coprime__crossproduct__nat,axiom,
! [A: nat,D: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),D)
=> ( aa_nat_o(aa_nat_fun_nat_o(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_10160_coprime__Suc__right__nat,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,Nb)) ).
% coprime_Suc_right_nat
tff(fact_10161_coprime__Suc__left__nat,axiom,
! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Nb)),Nb) ).
% coprime_Suc_left_nat
tff(fact_10162_coprime__common__divisor__nat,axiom,
! [A: nat,B: nat,Xb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),B)
=> ( Xb = one_one_nat ) ) ) ) ).
% coprime_common_divisor_nat
tff(fact_10163_Rats__no__bot__less,axiom,
! [Xb: real] :
? [X4: real] :
( aa_set_real_o(member_real(X4),field_5140801741446780682s_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Xb) ) ).
% Rats_no_bot_less
tff(fact_10164_Rats__dense__in__real,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
=> ? [X4: real] :
( aa_set_real_o(member_real(X4),field_5140801741446780682s_real)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X4)
& aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y) ) ) ).
% Rats_dense_in_real
tff(fact_10165_coprime__diff__one__right__nat,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ).
% coprime_diff_one_right_nat
tff(fact_10166_coprime__diff__one__left__nat,axiom,
! [Nb: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
=> aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),Nb) ) ).
% coprime_diff_one_left_nat
tff(fact_10167_sort__upt,axiom,
! [Mb: nat,Nb: nat] : linord738340561235409698at_nat(aTP_Lamm_aj,upt(Mb,Nb)) = upt(Mb,Nb) ).
% sort_upt
tff(fact_10168_sort__upto,axiom,
! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_ak,upto(I,J)) = upto(I,J) ).
% sort_upto
tff(fact_10169_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
tff(fact_10170_Rat_Opositive__def,axiom,
positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_vi) ).
% Rat.positive_def
tff(fact_10171_vanishes__mult__bounded,axiom,
! [X6: fun_nat_rat,Y6: fun_nat_rat] :
( ? [A7: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A7)
& ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N))),A7) )
=> ( vanishes(Y6)
=> vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vj(X6),Y6)) ) ) ).
% vanishes_mult_bounded
tff(fact_10172_vanishes__def,axiom,
! [X6: fun_nat_rat] :
( vanishes(X6)
<=> ! [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
=> ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R5) ) ) ) ).
% vanishes_def
tff(fact_10173_vanishesI,axiom,
! [X6: fun_nat_rat] :
( ! [R3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
=> ? [K4: nat] :
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N))),R3) ) )
=> vanishes(X6) ) ).
% vanishesI
tff(fact_10174_vanishesD,axiom,
! [X6: fun_nat_rat,R2: rat] :
( vanishes(X6)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
=> ? [K3: nat] :
! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),R2) ) ) ) ).
% vanishesD
tff(fact_10175_and__not__num_Opelims,axiom,
! [Xb: num,Xaa: num,Y: option_num] :
( ( bit_and_not_num(Xb,Xaa) = Y )
=> ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(Xb,Xaa))
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( ( Y = none_num )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,one)) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = some_num(one) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = none_num )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ( ( Xaa = one )
=> ( ( Y = some_num(aa_num_num(bit0,M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M2),one)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ( ( Xaa = one )
=> ( ( Y = some_num(aa_num_num(bit0,M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M2),one)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_sw,bit_and_not_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))) ) ) )
=> ~ ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
tff(fact_10176_and__num_Opelims,axiom,
! [Xb: num,Xaa: num,Y: option_num] :
( ( bit_un7362597486090784418nd_num(Xb,Xaa) = Y )
=> ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(Xb,Xaa))
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( ( Y = some_num(one) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,one)) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = none_num )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = some_num(one) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ( ( Xaa = one )
=> ( ( Y = none_num )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M2),one)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ( ( Xaa = one )
=> ( ( Y = some_num(one) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M2),one)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))) ) ) )
=> ~ ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_sw,bit_un7362597486090784418nd_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
tff(fact_10177_xor__num_Opelims,axiom,
! [Xb: num,Xaa: num,Y: option_num] :
( ( bit_un2480387367778600638or_num(Xb,Xaa) = Y )
=> ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(Xb,Xaa))
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( ( Y = none_num )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,one)) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = some_num(aa_num_num(bit1,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
=> ( ( ( Xb = one )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = some_num(aa_num_num(bit0,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ( ( Xaa = one )
=> ( ( Y = some_num(aa_num_num(bit1,M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M2),one)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit0,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M2,N))) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ( ( Xaa = one )
=> ( ( Y = some_num(aa_num_num(bit0,M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M2),one)) ) ) )
=> ( ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit0,N) )
=> ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M2,N))) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))) ) ) )
=> ~ ! [M2: num] :
( ( Xb = aa_num_num(bit1,M2) )
=> ! [N: num] :
( ( Xaa = aa_num_num(bit1,N) )
=> ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M2,N)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
tff(fact_10178_or__not__num__neg_Opelims,axiom,
! [Xb: num,Xaa: num,Y: num] :
( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
=> ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(Xb,Xaa))
=> ( ( ( Xb = one )
=> ( ( Xaa = one )
=> ( ( Y = one )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,one)) ) ) )
=> ( ( ( Xb = one )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit0,M2) )
=> ( ( Y = aa_num_num(bit1,M2) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit0,M2))) ) ) )
=> ( ( ( Xb = one )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit1,M2) )
=> ( ( Y = aa_num_num(bit1,M2) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit1,M2))) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit0,N) )
=> ( ( Xaa = one )
=> ( ( Y = aa_num_num(bit0,one) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),one)) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit0,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit0,M2) )
=> ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit0,M2))) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit0,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit1,M2) )
=> ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit1,M2))) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit1,N) )
=> ( ( Xaa = one )
=> ( ( Y = one )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),one)) ) ) )
=> ( ! [N: num] :
( ( Xb = aa_num_num(bit1,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit0,M2) )
=> ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit0,M2))) ) ) )
=> ~ ! [N: num] :
( ( Xb = aa_num_num(bit1,N) )
=> ! [M2: num] :
( ( Xaa = aa_num_num(bit1,M2) )
=> ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) )
=> ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit1,M2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
tff(fact_10179_cauchyD,axiom,
! [X6: fun_nat_rat,R2: rat] :
( cauchy(X6)
=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
=> ? [K3: nat] :
! [M3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M3)
=> ! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M3)),aa_nat_rat(X6,N5)))),R2) ) ) ) ) ).
% cauchyD
tff(fact_10180_cauchy__mult,axiom,
! [X6: fun_nat_rat,Y6: fun_nat_rat] :
( cauchy(X6)
=> ( cauchy(Y6)
=> cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_vj(X6),Y6)) ) ) ).
% cauchy_mult
tff(fact_10181_cauchy__imp__bounded,axiom,
! [X6: fun_nat_rat] :
( cauchy(X6)
=> ? [B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
& ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),B4) ) ) ).
% cauchy_imp_bounded
tff(fact_10182_cauchy__not__vanishes__cases,axiom,
! [X6: fun_nat_rat] :
( cauchy(X6)
=> ( ~ vanishes(X6)
=> ? [B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
& ? [K3: nat] :
( ! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X6,N5))) )
| ! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_nat_rat(X6,N5)) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
tff(fact_10183_cauchy__not__vanishes,axiom,
! [X6: fun_nat_rat] :
( cauchy(X6)
=> ( ~ vanishes(X6)
=> ? [B4: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
& ? [K3: nat] :
! [N5: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N5)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))) ) ) ) ) ).
% cauchy_not_vanishes
tff(fact_10184_cauchy__def,axiom,
! [X6: fun_nat_rat] :
( cauchy(X6)
<=> ! [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
=> ? [K2: nat] :
! [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M4)
=> ! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M4)),aa_nat_rat(X6,N3)))),R5) ) ) ) ) ).
% cauchy_def
tff(fact_10185_cauchyI,axiom,
! [X6: fun_nat_rat] :
( ! [R3: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
=> ? [K4: nat] :
! [M2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),M2)
=> ! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M2)),aa_nat_rat(X6,N)))),R3) ) ) )
=> cauchy(X6) ) ).
% cauchyI
tff(fact_10186_le__Real,axiom,
! [X6: fun_nat_rat,Y6: fun_nat_rat] :
( cauchy(X6)
=> ( cauchy(Y6)
=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real2(X6)),real2(Y6))
<=> ! [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
=> ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_rat_rat(plus_plus_rat(aa_nat_rat(Y6,N3)),R5)) ) ) ) ) ) ).
% le_Real
tff(fact_10187_mult__Real,axiom,
! [X6: fun_nat_rat,Y6: fun_nat_rat] :
( cauchy(X6)
=> ( cauchy(Y6)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,real2(X6)),real2(Y6)) = real2(aa_fun765758790203053448at_rat(aTP_Lamm_vj(X6),Y6)) ) ) ) ).
% mult_Real
tff(fact_10188_not__positive__Real,axiom,
! [X6: fun_nat_rat] :
( cauchy(X6)
=> ( ~ aa_real_o(positive2,real2(X6))
<=> ! [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
=> ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N3)),R5) ) ) ) ) ).
% not_positive_Real
tff(fact_10189_positive__Real,axiom,
! [X6: fun_nat_rat] :
( cauchy(X6)
=> ( aa_real_o(positive2,real2(X6))
<=> ? [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
& ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(X6,N3)) ) ) ) ) ).
% positive_Real
tff(fact_10190_Real_Opositive__mult,axiom,
! [Xb: real,Y: real] :
( aa_real_o(positive2,Xb)
=> ( aa_real_o(positive2,Y)
=> aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) ) ) ).
% Real.positive_mult
tff(fact_10191_less__real__def,axiom,
! [Xb: real,Y: real] :
( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
<=> aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)) ) ).
% less_real_def
tff(fact_10192_Real_Opositive_Orep__eq,axiom,
! [Xb: real] :
( aa_real_o(positive2,Xb)
<=> ? [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
& ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,Xb),N3)) ) ) ) ).
% Real.positive.rep_eq
tff(fact_10193_le__enumerate,axiom,
! [S3: set_nat,Nb: nat] :
( ~ aa_set_nat_o(finite_finite_nat,S3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S3),Nb)) ) ).
% le_enumerate
tff(fact_10194_finite__le__enumerate,axiom,
! [S3: set_nat,Nb: nat] :
( aa_set_nat_o(finite_finite_nat,S3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),finite_card_nat(S3))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S3),Nb)) ) ) ).
% finite_le_enumerate
tff(fact_10195_Real_Opositive__def,axiom,
positive2 = map_fu6719675619867407465at_o_o(rep_real,id_o,aTP_Lamm_vk) ).
% Real.positive_def
tff(fact_10196_Least__eq__0,axiom,
! [P: fun_nat_o] :
( aa_nat_o(P,zero_zero_nat)
=> ( ord_Least_nat(P) = zero_zero_nat ) ) ).
% Least_eq_0
tff(fact_10197_Least__Suc,axiom,
! [P: fun_nat_o,Nb: nat] :
( aa_nat_o(P,Nb)
=> ( ~ aa_nat_o(P,zero_zero_nat)
=> ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_ur(P))) ) ) ) ).
% Least_Suc
tff(fact_10198_Least__Suc2,axiom,
! [P: fun_nat_o,Nb: nat,Q: fun_nat_o,Mb: nat] :
( aa_nat_o(P,Nb)
=> ( aa_nat_o(Q,Mb)
=> ( ~ aa_nat_o(P,zero_zero_nat)
=> ( ! [K3: nat] :
( aa_nat_o(P,aa_nat_nat(suc,K3))
<=> aa_nat_o(Q,K3) )
=> ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).
% Least_Suc2
tff(fact_10199_DeMoivre2,axiom,
! [R2: real,A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(rcis(R2),A)),Nb) = aa_real_complex(rcis(aa_nat_real(aa_real_fun_nat_real(power_power_real,R2),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).
% DeMoivre2
tff(fact_10200_Re__rcis,axiom,
! [R2: real,A: real] : aa_complex_real(re,aa_real_complex(rcis(R2),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(cos_real,A)) ).
% Re_rcis
tff(fact_10201_Im__rcis,axiom,
! [R2: real,A: real] : aa_complex_real(im,aa_real_complex(rcis(R2),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(sin_real,A)) ).
% Im_rcis
tff(fact_10202_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_10203_rcis__def,axiom,
! [R2: real,A: real] : aa_real_complex(rcis(R2),A) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(cis,A)) ).
% rcis_def
tff(fact_10204_uniformity__real__def,axiom,
topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_vm,set_or5849166863359141190n_real(zero_zero_real))) ).
% uniformity_real_def
tff(fact_10205_uniformity__complex__def,axiom,
topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_vo,set_or5849166863359141190n_real(zero_zero_real))) ).
% uniformity_complex_def
tff(fact_10206_eventually__prod__sequentially,axiom,
! [P: fun_Pr4796943543331071514_nat_o] :
( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
<=> ? [N8: nat] :
! [M4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M4)
=> ! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N3)
=> aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M4)) ) ) ) ).
% eventually_prod_sequentially
tff(fact_10207_last__upt,axiom,
! [I: nat,J: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
=> ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).
% last_upt
tff(fact_10208_Real_Opositive_Oabs__eq,axiom,
! [Xb: fun_nat_rat] :
( realrel(Xb,Xb)
=> ( aa_real_o(positive2,real2(Xb))
<=> ? [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
& ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Xb,N3)) ) ) ) ) ).
% Real.positive.abs_eq
tff(fact_10209_ATP_Olambda__1,axiom,
! [Uu2: nat] : aa_nat_real(aTP_Lamm_ig,Uu2) = 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)),Uu2)),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,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).
% ATP.lambda_1
tff(fact_10210_ATP_Olambda__2,axiom,
! [Uu2: nat] : aa_nat_real(aTP_Lamm_li,Uu2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu2)) ).
% ATP.lambda_2
tff(fact_10211_ATP_Olambda__3,axiom,
! [Uu2: real] :
( aa_real_o(aTP_Lamm_qz,Uu2)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu2)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
& ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).
% ATP.lambda_3
tff(fact_10212_ATP_Olambda__4,axiom,
! [Uu2: nat] : aa_nat_real(aTP_Lamm_kl,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu2)) ).
% ATP.lambda_4
tff(fact_10213_ATP_Olambda__5,axiom,
! [Uu2: real] : aa_real_real(aTP_Lamm_tz,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).
% ATP.lambda_5
tff(fact_10214_ATP_Olambda__6,axiom,
! [Uu2: nat] : aa_nat_nat(aTP_Lamm_ux,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ).
% ATP.lambda_6
tff(fact_10215_ATP_Olambda__7,axiom,
! [Uu2: complex] : aa_complex_complex(aTP_Lamm_ma,Uu2) = aa_complex_complex(plus_plus_complex(Uu2),one_one_complex) ).
% ATP.lambda_7
tff(fact_10216_ATP_Olambda__8,axiom,
! [Uu2: real] : aa_real_real(aTP_Lamm_md,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),one_one_real) ).
% ATP.lambda_8
tff(fact_10217_ATP_Olambda__9,axiom,
! [Uu2: rat] : aa_rat_rat(aTP_Lamm_mb,Uu2) = aa_rat_rat(plus_plus_rat(Uu2),one_one_rat) ).
% ATP.lambda_9
tff(fact_10218_ATP_Olambda__10,axiom,
! [Uu2: nat] : aa_nat_nat(aTP_Lamm_me,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),one_one_nat) ).
% ATP.lambda_10
tff(fact_10219_ATP_Olambda__11,axiom,
! [Uu2: int] : aa_int_int(aTP_Lamm_mc,Uu2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),one_one_int) ).
% ATP.lambda_11
tff(fact_10220_ATP_Olambda__12,axiom,
! [Uu2: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aTP_Lamm_vi,Uu2)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_snd_int_int(Uu2))) ) ).
% ATP.lambda_12
tff(fact_10221_ATP_Olambda__13,axiom,
! [Uu2: real] : aa_real_real(aTP_Lamm_ts,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_if,Uu2)) ).
% ATP.lambda_13
tff(fact_10222_ATP_Olambda__14,axiom,
! [Uu2: real] : aa_rea5117059003799598681omplex(aTP_Lamm_vo,Uu2) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_vn(Uu2)))) ).
% ATP.lambda_14
tff(fact_10223_ATP_Olambda__15,axiom,
! [Uu2: real] : aa_rea5428163979925208277l_real(aTP_Lamm_vm,Uu2) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_vl(Uu2)))) ).
% ATP.lambda_15
tff(fact_10224_ATP_Olambda__16,axiom,
! [Uu2: nat] : aa_nat_real(aTP_Lamm_ue,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).
% ATP.lambda_16
tff(fact_10225_ATP_Olambda__17,axiom,
! [Uu2: num] : aa_num_option_num(aTP_Lamm_sw,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).
% ATP.lambda_17
tff(fact_10226_ATP_Olambda__18,axiom,
! [Uu2: num] : aa_num_option_num(aTP_Lamm_sv,Uu2) = some_num(aa_num_num(bit0,Uu2)) ).
% ATP.lambda_18
tff(fact_10227_ATP_Olambda__19,axiom,
! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ee,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).
% ATP.lambda_19
tff(fact_10228_ATP_Olambda__20,axiom,
! [Uu2: fun_nat_rat] :
( aa_fun_nat_rat_o(aTP_Lamm_vk,Uu2)
<=> ? [R5: rat] :
( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
& ? [K2: nat] :
! [N3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Uu2,N3)) ) ) ) ).
% ATP.lambda_20
tff(fact_10229_ATP_Olambda__21,axiom,
! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_su,Uu2) = some_num(one) ).
% ATP.lambda_21
tff(fact_10230_ATP_Olambda__22,axiom,
! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tb(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_sz(Uua),aTP_Lamm_ta(Uua),Uu2) ).
% ATP.lambda_22
tff(fact_10231_ATP_Olambda__23,axiom,
! [Uu2: fun_nat_real,Uua: nat] :
aa_nat_real(aTP_Lamm_lj(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu2,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_23
tff(fact_10232_ATP_Olambda__24,axiom,
! [Uu2: int,Uua: int] :
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rd,Uu2),Uua) = $ite(Uu2 = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu2)),Uua)),aa_int_int(abs_abs_int,Uu2))) ).
% ATP.lambda_24
tff(fact_10233_ATP_Olambda__25,axiom,
! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_tc,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_tb(Uua),Uu2) ).
% ATP.lambda_25
tff(fact_10234_ATP_Olambda__26,axiom,
! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_sz(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sv,bit_take_bit_num(Uu2,Uua)) ).
% ATP.lambda_26
tff(fact_10235_ATP_Olambda__27,axiom,
! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_sx(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sv,bit_take_bit_num(Uua,Uu2)) ).
% ATP.lambda_27
tff(fact_10236_ATP_Olambda__28,axiom,
! [Uu2: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qq(Uu2),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qp(Uu2),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).
% ATP.lambda_28
tff(fact_10237_ATP_Olambda__29,axiom,
! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qo(Uu2),Uua) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qn(Uu2),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).
% ATP.lambda_29
tff(fact_10238_ATP_Olambda__30,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ie,Uu2),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,Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ).
% ATP.lambda_30
tff(fact_10239_ATP_Olambda__31,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kv,Uu2),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,Uu2),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_31
tff(fact_10240_ATP_Olambda__32,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ll,Uu2),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,Uu2),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_32
tff(fact_10241_ATP_Olambda__33,axiom,
! [Uu2: real,Uua: real] :
( aa_real_o(aa_real_fun_real_o(aTP_Lamm_qy,Uu2),Uua)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).
% ATP.lambda_33
tff(fact_10242_ATP_Olambda__34,axiom,
! [Uu2: real,Uua: real] :
( aa_real_o(aa_real_fun_real_o(aTP_Lamm_qx,Uu2),Uua)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
& ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).
% ATP.lambda_34
tff(fact_10243_ATP_Olambda__35,axiom,
! [Uu2: code_integer,Uua: code_integer] :
aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_ru,Uu2),Uua) = $let(
l2: int,
l2:= 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(Uu2)),
$ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).
% ATP.lambda_35
tff(fact_10244_ATP_Olambda__36,axiom,
! [Uu2: nat,Uua: nat] :
aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_hy,Uu2),Uua) = $let(
m3: extended_enat,
m3:= times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu2)),
$ite(Uua = zero_zero_nat,m3,plus_p3455044024723400733d_enat(m3,one_on7984719198319812577d_enat)) ) ).
% ATP.lambda_36
tff(fact_10245_ATP_Olambda__37,axiom,
! [Uu2: nat,Uua: nat] :
aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hz,Uu2),Uua) = $let(
m3: complex,
m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu2)),
$ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).
% ATP.lambda_37
tff(fact_10246_ATP_Olambda__38,axiom,
! [Uu2: nat,Uua: nat] :
aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ib,Uu2),Uua) = $let(
m3: real,
m3:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu2)),
$ite(Uua = zero_zero_nat,m3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m3),one_one_real)) ) ).
% ATP.lambda_38
tff(fact_10247_ATP_Olambda__39,axiom,
! [Uu2: nat,Uua: nat] :
aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_hx,Uu2),Uua) = $let(
m3: rat,
m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu2)),
$ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).
% ATP.lambda_39
tff(fact_10248_ATP_Olambda__40,axiom,
! [Uu2: nat,Uua: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ic,Uu2),Uua) = $let(
m3: nat,
m3:= 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,Uu2)),
$ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).
% ATP.lambda_40
tff(fact_10249_ATP_Olambda__41,axiom,
! [Uu2: nat,Uua: nat] :
aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ia,Uu2),Uua) = $let(
m3: int,
m3:= 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,Uu2)),
$ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).
% ATP.lambda_41
tff(fact_10250_ATP_Olambda__42,axiom,
! [Uu2: complex,Uua: real] :
( aa_real_o(aTP_Lamm_rl(Uu2),Uua)
<=> ( ( aa_complex_complex(sgn_sgn_complex,Uu2) = aa_real_complex(cis,Uua) )
& aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
& aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).
% ATP.lambda_42
tff(fact_10251_ATP_Olambda__43,axiom,
! [Uu2: real,Uua: int] :
( aa_int_o(aTP_Lamm_ra(Uu2),Uua)
<=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu2)
& aa_real_o(aa_real_fun_real_o(ord_less_real,Uu2),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_43
tff(fact_10252_ATP_Olambda__44,axiom,
! [Uu2: rat,Uua: int] :
( aa_int_o(aTP_Lamm_rc(Uu2),Uua)
<=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu2)
& aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu2),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_44
tff(fact_10253_ATP_Olambda__45,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_if,Uu2),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,Uu2),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_45
tff(fact_10254_ATP_Olambda__46,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tt,Uu2),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,Uu2),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_46
tff(fact_10255_ATP_Olambda__47,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ua(Uu2),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(Uu2,Uua)) ).
% ATP.lambda_47
tff(fact_10256_ATP_Olambda__48,axiom,
! [Uu2: rat,Uua: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aTP_Lamm_vh(Uu2),Uua)
<=> ( ( Uu2 = aa_int_rat(fract(product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Uua))
& aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua)) ) ) ).
% ATP.lambda_48
tff(fact_10257_ATP_Olambda__49,axiom,
! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mf(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu2),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu2),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).
% ATP.lambda_49
tff(fact_10258_ATP_Olambda__50,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mh,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu2),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).
% ATP.lambda_50
tff(fact_10259_ATP_Olambda__51,axiom,
! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mg(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu2),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_51
tff(fact_10260_ATP_Olambda__52,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qs,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% ATP.lambda_52
tff(fact_10261_ATP_Olambda__53,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kn,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).
% ATP.lambda_53
tff(fact_10262_ATP_Olambda__54,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uv,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uua)),Uua) ).
% ATP.lambda_54
tff(fact_10263_ATP_Olambda__55,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uu2) ).
% ATP.lambda_55
tff(fact_10264_ATP_Olambda__56,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uua) ).
% ATP.lambda_56
tff(fact_10265_ATP_Olambda__57,axiom,
! [Uu2: nat,Uua: complex] :
( aa_complex_o(aTP_Lamm_bp(Uu2),Uua)
<=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).
% ATP.lambda_57
tff(fact_10266_ATP_Olambda__58,axiom,
! [Uu2: nat,Uua: real] :
( aa_real_o(aTP_Lamm_bo(Uu2),Uua)
<=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) = one_one_real ) ) ).
% ATP.lambda_58
tff(fact_10267_ATP_Olambda__59,axiom,
! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_up,Uu2),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,Uu2),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).
% ATP.lambda_59
tff(fact_10268_ATP_Olambda__60,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_id,Uu2),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,Uu2),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_60
tff(fact_10269_ATP_Olambda__61,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pq(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,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(Uu2,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_61
tff(fact_10270_ATP_Olambda__62,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pr(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,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(Uu2,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_62
tff(fact_10271_ATP_Olambda__63,axiom,
! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ps(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,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(Uu2,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_63
tff(fact_10272_ATP_Olambda__64,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pt(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,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(Uu2,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_64
tff(fact_10273_ATP_Olambda__65,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hf(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,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(Uu2,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_65
tff(fact_10274_ATP_Olambda__66,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hc(Uu2),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu2,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(Uu2,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_66
tff(fact_10275_ATP_Olambda__67,axiom,
! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_he(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,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(Uu2,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_67
tff(fact_10276_ATP_Olambda__68,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hd(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,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(Uu2,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_68
tff(fact_10277_ATP_Olambda__69,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gu(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,aa_nat_nat(suc,Uua))),aa_nat_real(Uu2,Uua)) ).
% ATP.lambda_69
tff(fact_10278_ATP_Olambda__70,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gs(Uu2),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu2,Uua)) ).
% ATP.lambda_70
tff(fact_10279_ATP_Olambda__71,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gt(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,aa_nat_nat(suc,Uua))),aa_nat_int(Uu2,Uua)) ).
% ATP.lambda_71
tff(fact_10280_ATP_Olambda__72,axiom,
! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_fv(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).
% ATP.lambda_72
tff(fact_10281_ATP_Olambda__73,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fx(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).
% ATP.lambda_73
tff(fact_10282_ATP_Olambda__74,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_fw(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).
% ATP.lambda_74
tff(fact_10283_ATP_Olambda__75,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_iy(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).
% ATP.lambda_75
tff(fact_10284_ATP_Olambda__76,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hb(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).
% ATP.lambda_76
tff(fact_10285_ATP_Olambda__77,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gz(Uu2),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).
% ATP.lambda_77
tff(fact_10286_ATP_Olambda__78,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ha(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).
% ATP.lambda_78
tff(fact_10287_ATP_Olambda__79,axiom,
! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gv(Uu2),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_79
tff(fact_10288_ATP_Olambda__80,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gy(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_80
tff(fact_10289_ATP_Olambda__81,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gw(Uu2),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_81
tff(fact_10290_ATP_Olambda__82,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gx(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_82
tff(fact_10291_ATP_Olambda__83,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ji(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(suc,Uua))) ).
% ATP.lambda_83
tff(fact_10292_ATP_Olambda__84,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jg(Uu2),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))) ).
% ATP.lambda_84
tff(fact_10293_ATP_Olambda__85,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jh(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(suc,Uua))) ).
% ATP.lambda_85
tff(fact_10294_ATP_Olambda__86,axiom,
! [Uu2: fun_complex_complex,Uua: complex] :
( aa_complex_o(aTP_Lamm_ol(Uu2),Uua)
<=> ( aa_complex_complex(Uu2,Uua) = one_one_complex ) ) ).
% ATP.lambda_86
tff(fact_10295_ATP_Olambda__87,axiom,
! [Uu2: fun_complex_real,Uua: complex] :
( aa_complex_o(aTP_Lamm_on(Uu2),Uua)
<=> ( aa_complex_real(Uu2,Uua) = one_one_real ) ) ).
% ATP.lambda_87
tff(fact_10296_ATP_Olambda__88,axiom,
! [Uu2: fun_complex_rat,Uua: complex] :
( aa_complex_o(aTP_Lamm_op(Uu2),Uua)
<=> ( aa_complex_rat(Uu2,Uua) = one_one_rat ) ) ).
% ATP.lambda_88
tff(fact_10297_ATP_Olambda__89,axiom,
! [Uu2: fun_complex_nat,Uua: complex] :
( aa_complex_o(aTP_Lamm_or(Uu2),Uua)
<=> ( aa_complex_nat(Uu2,Uua) = one_one_nat ) ) ).
% ATP.lambda_89
tff(fact_10298_ATP_Olambda__90,axiom,
! [Uu2: fun_complex_int,Uua: complex] :
( aa_complex_o(aTP_Lamm_os(Uu2),Uua)
<=> ( aa_complex_int(Uu2,Uua) = one_one_int ) ) ).
% ATP.lambda_90
tff(fact_10299_ATP_Olambda__91,axiom,
! [Uu2: fun_nat_complex,Uua: nat] :
( aa_nat_o(aTP_Lamm_ot(Uu2),Uua)
<=> ( aa_nat_complex(Uu2,Uua) = one_one_complex ) ) ).
% ATP.lambda_91
tff(fact_10300_ATP_Olambda__92,axiom,
! [Uu2: fun_int_complex,Uua: int] :
( aa_int_o(aTP_Lamm_ok(Uu2),Uua)
<=> ( aa_int_complex(Uu2,Uua) = one_one_complex ) ) ).
% ATP.lambda_92
tff(fact_10301_ATP_Olambda__93,axiom,
! [Uu2: fun_int_real,Uua: int] :
( aa_int_o(aTP_Lamm_om(Uu2),Uua)
<=> ( aa_int_real(Uu2,Uua) = one_one_real ) ) ).
% ATP.lambda_93
tff(fact_10302_ATP_Olambda__94,axiom,
! [Uu2: fun_int_rat,Uua: int] :
( aa_int_o(aTP_Lamm_oo(Uu2),Uua)
<=> ( aa_int_rat(Uu2,Uua) = one_one_rat ) ) ).
% ATP.lambda_94
tff(fact_10303_ATP_Olambda__95,axiom,
! [Uu2: fun_int_nat,Uua: int] :
( aa_int_o(aTP_Lamm_oq(Uu2),Uua)
<=> ( aa_int_nat(Uu2,Uua) = one_one_nat ) ) ).
% ATP.lambda_95
tff(fact_10304_ATP_Olambda__96,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uo(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_ua(Uu2),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_96
tff(fact_10305_ATP_Olambda__97,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_un(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_ua(Uu2),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_97
tff(fact_10306_ATP_Olambda__98,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vg,Uu2),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,Uu2),Uua))),Uu2) ).
% ATP.lambda_98
tff(fact_10307_ATP_Olambda__99,axiom,
! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kx(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua)) ).
% ATP.lambda_99
tff(fact_10308_ATP_Olambda__100,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kw,Uu2),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,Uu2),Uua)) ).
% ATP.lambda_100
tff(fact_10309_ATP_Olambda__101,axiom,
! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_st(Uu2),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu2)),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,Uu2))),aa_num_int(numeral_numeral_int,Uua))) ).
% ATP.lambda_101
tff(fact_10310_ATP_Olambda__102,axiom,
! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_tg,Uu2),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)),Uu2)) ).
% ATP.lambda_102
tff(fact_10311_ATP_Olambda__103,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ku,Uu2),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,Uu2),Uua)) ).
% ATP.lambda_103
tff(fact_10312_ATP_Olambda__104,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_km,Uu2),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,Uu2),Uua)) ).
% ATP.lambda_104
tff(fact_10313_ATP_Olambda__105,axiom,
! [Uu2: code_integer,Uua: code_integer] :
aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_rv,Uu2),Uua) = $let(
l2: num,
l2:= code_num_of_integer(Uu2),
$let(
l3: num,
l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
$ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).
% ATP.lambda_105
tff(fact_10314_ATP_Olambda__106,axiom,
! [Uu2: code_integer,Uua: code_integer] :
aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_rw,Uu2),Uua) = $let(
l2: nat,
l2:= code_nat_of_integer(Uu2),
$let(
l3: nat,
l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
$ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).
% ATP.lambda_106
tff(fact_10315_ATP_Olambda__107,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pz(Uu2),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu2) ).
% ATP.lambda_107
tff(fact_10316_ATP_Olambda__108,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qb,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu2) ).
% ATP.lambda_108
tff(fact_10317_ATP_Olambda__109,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_qa(Uu2),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu2) ).
% ATP.lambda_109
tff(fact_10318_ATP_Olambda__110,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_de,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),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_110
tff(fact_10319_ATP_Olambda__111,axiom,
! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_df,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),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_111
tff(fact_10320_ATP_Olambda__112,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dg,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),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_112
tff(fact_10321_ATP_Olambda__113,axiom,
! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dh,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),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_113
tff(fact_10322_ATP_Olambda__114,axiom,
! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ls(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).
% ATP.lambda_114
tff(fact_10323_ATP_Olambda__115,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lu,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).
% ATP.lambda_115
tff(fact_10324_ATP_Olambda__116,axiom,
! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lt(Uu2),Uua) = aa_rat_rat(plus_plus_rat(Uu2),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).
% ATP.lambda_116
tff(fact_10325_ATP_Olambda__117,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qt,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)) ).
% ATP.lambda_117
tff(fact_10326_ATP_Olambda__118,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),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_118
tff(fact_10327_ATP_Olambda__119,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ui,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).
% ATP.lambda_119
tff(fact_10328_ATP_Olambda__120,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uf,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).
% ATP.lambda_120
tff(fact_10329_ATP_Olambda__121,axiom,
! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ow(Uu2),Uua) = aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).
% ATP.lambda_121
tff(fact_10330_ATP_Olambda__122,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oy,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).
% ATP.lambda_122
tff(fact_10331_ATP_Olambda__123,axiom,
! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ox(Uu2),Uua) = aa_rat_rat(minus_minus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).
% ATP.lambda_123
tff(fact_10332_ATP_Olambda__124,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oz,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).
% ATP.lambda_124
tff(fact_10333_ATP_Olambda__125,axiom,
! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_pa(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).
% ATP.lambda_125
tff(fact_10334_ATP_Olambda__126,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).
% ATP.lambda_126
tff(fact_10335_ATP_Olambda__127,axiom,
! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pi(Uu2),Uua) = aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).
% ATP.lambda_127
tff(fact_10336_ATP_Olambda__128,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pk,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).
% ATP.lambda_128
tff(fact_10337_ATP_Olambda__129,axiom,
! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_pl(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).
% ATP.lambda_129
tff(fact_10338_ATP_Olambda__130,axiom,
! [Uu2: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vb,Uu2),Uua)
<=> ( Uua = aa_nat_nat(suc,Uu2) ) ) ).
% ATP.lambda_130
tff(fact_10339_ATP_Olambda__131,axiom,
! [Uu2: real,Uua: real] :
( aa_real_o(aa_real_fun_real_o(aTP_Lamm_qh,Uu2),Uua)
<=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu2) ) ).
% ATP.lambda_131
tff(fact_10340_ATP_Olambda__132,axiom,
! [Uu2: set_nat,Uua: set_nat] :
( aa_set_nat_o(aTP_Lamm_qd(Uu2),Uua)
<=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu2) ) ).
% ATP.lambda_132
tff(fact_10341_ATP_Olambda__133,axiom,
! [Uu2: rat,Uua: rat] :
( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_qe,Uu2),Uua)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu2) ) ).
% ATP.lambda_133
tff(fact_10342_ATP_Olambda__134,axiom,
! [Uu2: num,Uua: num] :
( aa_num_o(aa_num_fun_num_o(aTP_Lamm_qf,Uu2),Uua)
<=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu2) ) ).
% ATP.lambda_134
tff(fact_10343_ATP_Olambda__135,axiom,
! [Uu2: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bi,Uu2),Uua)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu2) ) ).
% ATP.lambda_135
tff(fact_10344_ATP_Olambda__136,axiom,
! [Uu2: int,Uua: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qg,Uu2),Uua)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu2) ) ).
% ATP.lambda_136
tff(fact_10345_ATP_Olambda__137,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tf,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu2) ).
% ATP.lambda_137
tff(fact_10346_ATP_Olambda__138,axiom,
! [Uu2: real,Uua: real] :
( aa_real_o(aa_real_fun_real_o(aTP_Lamm_it,Uu2),Uua)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu2) ) ).
% ATP.lambda_138
tff(fact_10347_ATP_Olambda__139,axiom,
! [Uu2: rat,Uua: rat] :
( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_iq,Uu2),Uua)
<=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu2) ) ).
% ATP.lambda_139
tff(fact_10348_ATP_Olambda__140,axiom,
! [Uu2: num,Uua: num] :
( aa_num_o(aa_num_fun_num_o(aTP_Lamm_ir,Uu2),Uua)
<=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu2) ) ).
% ATP.lambda_140
tff(fact_10349_ATP_Olambda__141,axiom,
! [Uu2: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bj,Uu2),Uua)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2) ) ).
% ATP.lambda_141
tff(fact_10350_ATP_Olambda__142,axiom,
! [Uu2: int,Uua: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_is,Uu2),Uua)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu2) ) ).
% ATP.lambda_142
tff(fact_10351_ATP_Olambda__143,axiom,
! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_as,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu2) ).
% ATP.lambda_143
tff(fact_10352_ATP_Olambda__144,axiom,
! [Uu2: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_at(Uu2),Uua) = aa_rat_rat(times_times_rat(Uua),Uu2) ).
% ATP.lambda_144
tff(fact_10353_ATP_Olambda__145,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_au,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ).
% ATP.lambda_145
tff(fact_10354_ATP_Olambda__146,axiom,
! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_av(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu2) ).
% ATP.lambda_146
tff(fact_10355_ATP_Olambda__147,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu2) ).
% ATP.lambda_147
tff(fact_10356_ATP_Olambda__148,axiom,
! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_th,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ).
% ATP.lambda_148
tff(fact_10357_ATP_Olambda__149,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ).
% ATP.lambda_149
tff(fact_10358_ATP_Olambda__150,axiom,
! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu2) ).
% ATP.lambda_150
tff(fact_10359_ATP_Olambda__151,axiom,
! [Uu2: code_integer,Uua: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ea,Uu2),Uua)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu2) ) ).
% ATP.lambda_151
tff(fact_10360_ATP_Olambda__152,axiom,
! [Uu2: complex,Uua: complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_dx,Uu2),Uua)
<=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Uua),Uu2) ) ).
% ATP.lambda_152
tff(fact_10361_ATP_Olambda__153,axiom,
! [Uu2: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dy,Uu2),Uua)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu2) ) ).
% ATP.lambda_153
tff(fact_10362_ATP_Olambda__154,axiom,
! [Uu2: int,Uua: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dz,Uu2),Uua)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu2) ) ).
% ATP.lambda_154
tff(fact_10363_ATP_Olambda__155,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_sk,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu2) ).
% ATP.lambda_155
tff(fact_10364_ATP_Olambda__156,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qc,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu2) ).
% ATP.lambda_156
tff(fact_10365_ATP_Olambda__157,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ud,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ).
% ATP.lambda_157
tff(fact_10366_ATP_Olambda__158,axiom,
! [Uu2: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aTP_Lamm_a(Uu2),Uua)
<=> member8440522571783428010at_nat(Uua,Uu2) ) ).
% ATP.lambda_158
tff(fact_10367_ATP_Olambda__159,axiom,
! [Uu2: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aTP_Lamm_ab(Uu2),Uua)
<=> member5262025264175285858nt_int(Uua,Uu2) ) ).
% ATP.lambda_159
tff(fact_10368_ATP_Olambda__160,axiom,
! [Uu2: set_complex,Uua: complex] :
( aa_complex_o(aTP_Lamm_ac(Uu2),Uua)
<=> aa_set_complex_o(member_complex(Uua),Uu2) ) ).
% ATP.lambda_160
tff(fact_10369_ATP_Olambda__161,axiom,
! [Uu2: set_real,Uua: real] :
( aa_real_o(aTP_Lamm_aa(Uu2),Uua)
<=> aa_set_real_o(member_real(Uua),Uu2) ) ).
% ATP.lambda_161
tff(fact_10370_ATP_Olambda__162,axiom,
! [Uu2: set_list_nat,Uua: list_nat] :
( aa_list_nat_o(aTP_Lamm_ad(Uu2),Uua)
<=> member_list_nat(Uua,Uu2) ) ).
% ATP.lambda_162
tff(fact_10371_ATP_Olambda__163,axiom,
! [Uu2: set_nat,Uua: nat] :
( aa_nat_o(aTP_Lamm_ae(Uu2),Uua)
<=> aa_set_nat_o(member_nat(Uua),Uu2) ) ).
% ATP.lambda_163
tff(fact_10372_ATP_Olambda__164,axiom,
! [Uu2: set_int,Uua: int] :
( aa_int_o(aTP_Lamm_af(Uu2),Uua)
<=> aa_set_int_o(member_int(Uua),Uu2) ) ).
% ATP.lambda_164
tff(fact_10373_ATP_Olambda__165,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_vf(Uu2),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu2) ).
% ATP.lambda_165
tff(fact_10374_ATP_Olambda__166,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kk(Uu2),Uua) = aa_nat_real(Uu2,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_166
tff(fact_10375_ATP_Olambda__167,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kj(Uu2),Uua) = aa_nat_real(Uu2,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_167
tff(fact_10376_ATP_Olambda__168,axiom,
! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_la(Uu2),Uua) = aa_nat_complex(Uu2,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_168
tff(fact_10377_ATP_Olambda__169,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gh(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_169
tff(fact_10378_ATP_Olambda__170,axiom,
! [Uu2: fun_nat_o,Uua: nat] :
( aa_nat_o(aTP_Lamm_ur(Uu2),Uua)
<=> aa_nat_o(Uu2,aa_nat_nat(suc,Uua)) ) ).
% ATP.lambda_170
tff(fact_10379_ATP_Olambda__171,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gq(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_171
tff(fact_10380_ATP_Olambda__172,axiom,
! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gg(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_172
tff(fact_10381_ATP_Olambda__173,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gr(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_173
tff(fact_10382_ATP_Olambda__174,axiom,
! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ta(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).
% ATP.lambda_174
tff(fact_10383_ATP_Olambda__175,axiom,
! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_sy(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).
% ATP.lambda_175
tff(fact_10384_ATP_Olambda__176,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ss,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sr(Uu2),Uua)) ).
% ATP.lambda_176
tff(fact_10385_ATP_Olambda__177,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sq,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sp(Uu2),Uua)) ).
% ATP.lambda_177
tff(fact_10386_ATP_Olambda__178,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_so,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sn(Uu2),Uua)) ).
% ATP.lambda_178
tff(fact_10387_ATP_Olambda__179,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_sm,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sl(Uu2),Uua)) ).
% ATP.lambda_179
tff(fact_10388_ATP_Olambda__180,axiom,
! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sj,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_si(Uu2),Uua)) ).
% ATP.lambda_180
tff(fact_10389_ATP_Olambda__181,axiom,
! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_tw(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_tv(Uu2),Uua)) ).
% ATP.lambda_181
tff(fact_10390_ATP_Olambda__182,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ri(Uu2),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).
% ATP.lambda_182
tff(fact_10391_ATP_Olambda__183,axiom,
! [Uu2: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ef,Uu2),Uua)
<=> ~ aa_nat_o(aa_nat_fun_nat_o(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,Uu2),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_183
tff(fact_10392_ATP_Olambda__184,axiom,
! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pf(Uu2),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).
% ATP.lambda_184
tff(fact_10393_ATP_Olambda__185,axiom,
! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ph(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).
% ATP.lambda_185
tff(fact_10394_ATP_Olambda__186,axiom,
! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_pg(Uu2),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_186
tff(fact_10395_ATP_Olambda__187,axiom,
! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_lv(Uu2),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).
% ATP.lambda_187
tff(fact_10396_ATP_Olambda__188,axiom,
! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ly(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).
% ATP.lambda_188
tff(fact_10397_ATP_Olambda__189,axiom,
! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_lw(Uu2),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_189
tff(fact_10398_ATP_Olambda__190,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_lz(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).
% ATP.lambda_190
tff(fact_10399_ATP_Olambda__191,axiom,
! [Uu2: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_lx(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).
% ATP.lambda_191
tff(fact_10400_ATP_Olambda__192,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_vc(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).
% ATP.lambda_192
tff(fact_10401_ATP_Olambda__193,axiom,
! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uh,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).
% ATP.lambda_193
tff(fact_10402_ATP_Olambda__194,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_te,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu2),Uua)) ).
% ATP.lambda_194
tff(fact_10403_ATP_Olambda__195,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_td,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu2)) ).
% ATP.lambda_195
tff(fact_10404_ATP_Olambda__196,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sc,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu2),Uua)) ).
% ATP.lambda_196
tff(fact_10405_ATP_Olambda__197,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sb,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu2)) ).
% ATP.lambda_197
tff(fact_10406_ATP_Olambda__198,axiom,
! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_jd(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu2,Uua)) ).
% ATP.lambda_198
tff(fact_10407_ATP_Olambda__199,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uc(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).
% ATP.lambda_199
tff(fact_10408_ATP_Olambda__200,axiom,
! [Uu2: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_nl(Uu2),Uua) = times_times_complex(aa_nat_complex(Uu2,Uua)) ).
% ATP.lambda_200
tff(fact_10409_ATP_Olambda__201,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_nm(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)) ).
% ATP.lambda_201
tff(fact_10410_ATP_Olambda__202,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_nn(Uu2),Uua) = times_times_rat(aa_nat_rat(Uu2,Uua)) ).
% ATP.lambda_202
tff(fact_10411_ATP_Olambda__203,axiom,
! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_no(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uua)) ).
% ATP.lambda_203
tff(fact_10412_ATP_Olambda__204,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_np(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)) ).
% ATP.lambda_204
tff(fact_10413_ATP_Olambda__205,axiom,
! [Uu2: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pu(Uu2),Uua) = plus_plus_complex(aa_nat_complex(Uu2,Uua)) ).
% ATP.lambda_205
tff(fact_10414_ATP_Olambda__206,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_py(Uu2),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uua)) ).
% ATP.lambda_206
tff(fact_10415_ATP_Olambda__207,axiom,
! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_pv(Uu2),Uua) = plus_plus_rat(aa_nat_rat(Uu2,Uua)) ).
% ATP.lambda_207
tff(fact_10416_ATP_Olambda__208,axiom,
! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_px(Uu2),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uua)) ).
% ATP.lambda_208
tff(fact_10417_ATP_Olambda__209,axiom,
! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_pw(Uu2),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uua)) ).
% ATP.lambda_209
tff(fact_10418_ATP_Olambda__210,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_eq(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).
% ATP.lambda_210
tff(fact_10419_ATP_Olambda__211,axiom,
! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ep(Uu2),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu2,Uua)) ).
% ATP.lambda_211
tff(fact_10420_ATP_Olambda__212,axiom,
! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_um(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).
% ATP.lambda_212
tff(fact_10421_ATP_Olambda__213,axiom,
! [Uu2: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_ty(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ).
% ATP.lambda_213
tff(fact_10422_ATP_Olambda__214,axiom,
! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_ve(Uu2),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Uu2)) ).
% ATP.lambda_214
tff(fact_10423_ATP_Olambda__215,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lk(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(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(Uu2,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_215
tff(fact_10424_ATP_Olambda__216,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ki(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu2,Uub),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_216
tff(fact_10425_ATP_Olambda__217,axiom,
! [Uu2: num,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ed(Uu2),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu2)),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(Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).
% ATP.lambda_217
tff(fact_10426_ATP_Olambda__218,axiom,
! [Uu2: num,Uua: nat,Uub: nat] :
aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_eb(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu2)),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,Uu2))),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_218
tff(fact_10427_ATP_Olambda__219,axiom,
! [Uu2: num,Uua: int,Uub: int] :
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ec(Uu2),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),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,Uu2))),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_219
tff(fact_10428_ATP_Olambda__220,axiom,
! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] :
aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_my(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_complex_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_220
tff(fact_10429_ATP_Olambda__221,axiom,
! [Uu2: complex,Uua: fun_complex_real,Uub: complex] :
aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_nc(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_complex_real(Uua,Uub),one_one_real) ).
% ATP.lambda_221
tff(fact_10430_ATP_Olambda__222,axiom,
! [Uu2: real,Uua: fun_real_complex,Uub: real] :
aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_mv(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_222
tff(fact_10431_ATP_Olambda__223,axiom,
! [Uu2: real,Uua: fun_real_real,Uub: real] :
aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mz(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_real(Uua,Uub),one_one_real) ).
% ATP.lambda_223
tff(fact_10432_ATP_Olambda__224,axiom,
! [Uu2: real,Uua: fun_real_rat,Uub: real] :
aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nd(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_224
tff(fact_10433_ATP_Olambda__225,axiom,
! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] :
aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mw(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_nat_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_225
tff(fact_10434_ATP_Olambda__226,axiom,
! [Uu2: nat,Uua: fun_nat_real,Uub: nat] :
aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_nat_real(Uua,Uub),one_one_real) ).
% ATP.lambda_226
tff(fact_10435_ATP_Olambda__227,axiom,
! [Uu2: nat,Uua: fun_nat_rat,Uub: nat] :
aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ne(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_227
tff(fact_10436_ATP_Olambda__228,axiom,
! [Uu2: int,Uua: fun_int_complex,Uub: int] :
aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_mx(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_228
tff(fact_10437_ATP_Olambda__229,axiom,
! [Uu2: int,Uua: fun_int_real,Uub: int] :
aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_nb(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_real(Uua,Uub),one_one_real) ).
% ATP.lambda_229
tff(fact_10438_ATP_Olambda__230,axiom,
! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] :
aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_mo(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_complex_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_230
tff(fact_10439_ATP_Olambda__231,axiom,
! [Uu2: complex,Uua: fun_complex_real,Uub: complex] :
aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ms(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_complex_real(Uua,Uub),one_one_real) ).
% ATP.lambda_231
tff(fact_10440_ATP_Olambda__232,axiom,
! [Uu2: real,Uua: fun_real_complex,Uub: real] :
aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ml(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_232
tff(fact_10441_ATP_Olambda__233,axiom,
! [Uu2: real,Uua: fun_real_real,Uub: real] :
aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mp(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_real(Uua,Uub),one_one_real) ).
% ATP.lambda_233
tff(fact_10442_ATP_Olambda__234,axiom,
! [Uu2: real,Uua: fun_real_rat,Uub: real] :
aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_mt(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_234
tff(fact_10443_ATP_Olambda__235,axiom,
! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] :
aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mm(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_nat_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_235
tff(fact_10444_ATP_Olambda__236,axiom,
! [Uu2: nat,Uua: fun_nat_real,Uub: nat] :
aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mq(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_nat_real(Uua,Uub),one_one_real) ).
% ATP.lambda_236
tff(fact_10445_ATP_Olambda__237,axiom,
! [Uu2: nat,Uua: fun_nat_rat,Uub: nat] :
aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mu(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_nat_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_237
tff(fact_10446_ATP_Olambda__238,axiom,
! [Uu2: int,Uua: fun_int_complex,Uub: int] :
aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_mn(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_complex(Uua,Uub),one_one_complex) ).
% ATP.lambda_238
tff(fact_10447_ATP_Olambda__239,axiom,
! [Uu2: int,Uua: fun_int_real,Uub: int] :
aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mr(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_real(Uua,Uub),one_one_real) ).
% ATP.lambda_239
tff(fact_10448_ATP_Olambda__240,axiom,
! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_sf(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu2),Uub))) ).
% ATP.lambda_240
tff(fact_10449_ATP_Olambda__241,axiom,
! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_sg(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu2),Uub))) ).
% ATP.lambda_241
tff(fact_10450_ATP_Olambda__242,axiom,
! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_se(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu2,Uub))) ).
% ATP.lambda_242
tff(fact_10451_ATP_Olambda__243,axiom,
! [Uu2: fun_complex_complex,Uua: fun_complex_o,Uub: complex] :
aa_complex_complex(aa_fun3681620399087435360omplex(aTP_Lamm_nx(Uu2),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_complex(Uu2,Uub),one_one_complex) ).
% ATP.lambda_243
tff(fact_10452_ATP_Olambda__244,axiom,
! [Uu2: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_ob(Uu2),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu2,Uub),one_one_real) ).
% ATP.lambda_244
tff(fact_10453_ATP_Olambda__245,axiom,
! [Uu2: fun_real_complex,Uua: fun_real_o,Uub: real] :
aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_nr(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu2,Uub),one_one_complex) ).
% ATP.lambda_245
tff(fact_10454_ATP_Olambda__246,axiom,
! [Uu2: fun_real_real,Uua: fun_real_o,Uub: real] :
aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_ny(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu2,Uub),one_one_real) ).
% ATP.lambda_246
tff(fact_10455_ATP_Olambda__247,axiom,
! [Uu2: fun_real_rat,Uua: fun_real_o,Uub: real] :
aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_oc(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu2,Uub),one_one_rat) ).
% ATP.lambda_247
tff(fact_10456_ATP_Olambda__248,axiom,
! [Uu2: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_nt(Uu2),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu2,Uub),one_one_complex) ).
% ATP.lambda_248
tff(fact_10457_ATP_Olambda__249,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_o,Uub: nat] :
aa_nat_real(aa_fun5510454223943551838t_real(aTP_Lamm_nz(Uu2),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_real(Uu2,Uub),one_one_real) ).
% ATP.lambda_249
tff(fact_10458_ATP_Olambda__250,axiom,
! [Uu2: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_od(Uu2),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu2,Uub),one_one_rat) ).
% ATP.lambda_250
tff(fact_10459_ATP_Olambda__251,axiom,
! [Uu2: fun_int_complex,Uua: fun_int_o,Uub: int] :
aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_nv(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu2,Uub),one_one_complex) ).
% ATP.lambda_251
tff(fact_10460_ATP_Olambda__252,axiom,
! [Uu2: fun_int_real,Uua: fun_int_o,Uub: int] :
aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_oa(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu2,Uub),one_one_real) ).
% ATP.lambda_252
tff(fact_10461_ATP_Olambda__253,axiom,
! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qn(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).
% ATP.lambda_253
tff(fact_10462_ATP_Olambda__254,axiom,
! [Uu2: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qp(Uu2),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).
% ATP.lambda_254
tff(fact_10463_ATP_Olambda__255,axiom,
! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_tx(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ).
% ATP.lambda_255
tff(fact_10464_ATP_Olambda__256,axiom,
! [Uu2: nat,Uua: complex,Uub: nat] :
aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ld(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu2,one_one_complex,zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_256
tff(fact_10465_ATP_Olambda__257,axiom,
! [Uu2: nat,Uua: real,Uub: nat] :
aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_le(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
$ite(Uub = Uu2,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_257
tff(fact_10466_ATP_Olambda__258,axiom,
! [Uu2: nat,Uua: int,Uub: nat] :
aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lf(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
$ite(Uub = Uu2,one_one_int,zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).
% ATP.lambda_258
tff(fact_10467_ATP_Olambda__259,axiom,
! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_sd(Uu2),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu2),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).
% ATP.lambda_259
tff(fact_10468_ATP_Olambda__260,axiom,
! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rr(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rq(Uua),Uub)),quotient_of(Uu2)) ).
% ATP.lambda_260
tff(fact_10469_ATP_Olambda__261,axiom,
! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rp(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ro(Uua),Uub)),quotient_of(Uu2)) ).
% ATP.lambda_261
tff(fact_10470_ATP_Olambda__262,axiom,
! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rn(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rm(Uua),Uub)),quotient_of(Uu2)) ).
% ATP.lambda_262
tff(fact_10471_ATP_Olambda__263,axiom,
! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rk(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rj(Uua),Uub)),quotient_of(Uu2)) ).
% ATP.lambda_263
tff(fact_10472_ATP_Olambda__264,axiom,
! [Uu2: rat,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rh(Uu2),Uua),Uub)
<=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_rg(Uua),Uub)),quotient_of(Uu2)) ) ).
% ATP.lambda_264
tff(fact_10473_ATP_Olambda__265,axiom,
! [Uu2: rat,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rf(Uu2),Uua),Uub)
<=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_re(Uua),Uub)),quotient_of(Uu2)) ) ).
% ATP.lambda_265
tff(fact_10474_ATP_Olambda__266,axiom,
! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tm(Uu2),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(Uu2,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_266
tff(fact_10475_ATP_Olambda__267,axiom,
! [Uu2: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_ko(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_267
tff(fact_10476_ATP_Olambda__268,axiom,
! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_kp(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_268
tff(fact_10477_ATP_Olambda__269,axiom,
! [Uu2: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_kq(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_269
tff(fact_10478_ATP_Olambda__270,axiom,
! [Uu2: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_kr(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_270
tff(fact_10479_ATP_Olambda__271,axiom,
! [Uu2: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_ks(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_271
tff(fact_10480_ATP_Olambda__272,axiom,
! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fy(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).
% ATP.lambda_272
tff(fact_10481_ATP_Olambda__273,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_273
tff(fact_10482_ATP_Olambda__274,axiom,
! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fz(Uu2),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).
% ATP.lambda_274
tff(fact_10483_ATP_Olambda__275,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tu(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,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_275
tff(fact_10484_ATP_Olambda__276,axiom,
! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kt(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_276
tff(fact_10485_ATP_Olambda__277,axiom,
! [Uu2: nat,Uua: nat,Uub: list_nat] :
( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_uz(Uu2),Uua),Uub)
<=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),one_one_nat) )
& ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_277
tff(fact_10486_ATP_Olambda__278,axiom,
! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ln(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu2,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_278
tff(fact_10487_ATP_Olambda__279,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lp(Uu2),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(Uu2,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_279
tff(fact_10488_ATP_Olambda__280,axiom,
! [Uu2: nat,Uua: nat,Uub: list_nat] :
( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_va(Uu2),Uua),Uub)
<=> ( ( size_size_list_nat(Uub) = Uu2 )
& ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).
% ATP.lambda_280
tff(fact_10489_ATP_Olambda__281,axiom,
! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_be(Uu2),Uua),Uub)
<=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).
% ATP.lambda_281
tff(fact_10490_ATP_Olambda__282,axiom,
! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bd(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).
% ATP.lambda_282
tff(fact_10491_ATP_Olambda__283,axiom,
! [Uu2: set_o,Uua: nat,Uub: list_o] :
( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bf(Uu2),Uua),Uub)
<=> ( ord_less_eq_set_o(set_o2(Uub),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).
% ATP.lambda_283
tff(fact_10492_ATP_Olambda__284,axiom,
! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bh(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).
% ATP.lambda_284
tff(fact_10493_ATP_Olambda__285,axiom,
! [Uu2: set_int,Uua: nat,Uub: list_int] :
( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bg(Uu2),Uua),Uub)
<=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).
% ATP.lambda_285
tff(fact_10494_ATP_Olambda__286,axiom,
! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_az(Uu2),Uua),Uub)
<=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu2)
& ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).
% ATP.lambda_286
tff(fact_10495_ATP_Olambda__287,axiom,
! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_ay(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2)
& ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).
% ATP.lambda_287
tff(fact_10496_ATP_Olambda__288,axiom,
! [Uu2: set_o,Uua: nat,Uub: list_o] :
( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_ba(Uu2),Uua),Uub)
<=> ( ord_less_eq_set_o(set_o2(Uub),Uu2)
& ( size_size_list_o(Uub) = Uua ) ) ) ).
% ATP.lambda_288
tff(fact_10497_ATP_Olambda__289,axiom,
! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bc(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu2)
& ( size_size_list_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_289
tff(fact_10498_ATP_Olambda__290,axiom,
! [Uu2: set_int,Uua: nat,Uub: list_int] :
( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bb(Uu2),Uua),Uub)
<=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu2)
& ( size_size_list_int(Uub) = Uua ) ) ) ).
% ATP.lambda_290
tff(fact_10499_ATP_Olambda__291,axiom,
! [Uu2: nat,Uua: nat,Uub: list_nat] :
( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_uy(Uu2),Uua),Uub)
<=> ( ( size_size_list_nat(Uub) = Uu2 )
& ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_291
tff(fact_10500_ATP_Olambda__292,axiom,
! [Uu2: set_nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sa(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).
% ATP.lambda_292
tff(fact_10501_ATP_Olambda__293,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vd(Uu2),Uua),Uub)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).
% ATP.lambda_293
tff(fact_10502_ATP_Olambda__294,axiom,
! [Uu2: set_nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rz(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).
% ATP.lambda_294
tff(fact_10503_ATP_Olambda__295,axiom,
! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu2),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_295
tff(fact_10504_ATP_Olambda__296,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lo(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu2),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_296
tff(fact_10505_ATP_Olambda__297,axiom,
! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu2),Uub)) ).
% ATP.lambda_297
tff(fact_10506_ATP_Olambda__298,axiom,
! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lq(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_298
tff(fact_10507_ATP_Olambda__299,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_299
tff(fact_10508_ATP_Olambda__300,axiom,
! [Uu2: int,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ej(Uu2),Uua),Uub)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uua)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).
% ATP.lambda_300
tff(fact_10509_ATP_Olambda__301,axiom,
! [Uu2: int,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ei(Uu2),Uua),Uub)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).
% ATP.lambda_301
tff(fact_10510_ATP_Olambda__302,axiom,
! [Uu2: int,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bl(Uu2),Uua),Uub)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).
% ATP.lambda_302
tff(fact_10511_ATP_Olambda__303,axiom,
! [Uu2: int,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bm(Uu2),Uua),Uub)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu2),Uub)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).
% ATP.lambda_303
tff(fact_10512_ATP_Olambda__304,axiom,
! [Uu2: int,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bk(Uu2),Uua),Uub)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu2),Uub)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).
% ATP.lambda_304
tff(fact_10513_ATP_Olambda__305,axiom,
! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_am(Uu2),Uua),Uub)
<=> ( aa_nat_o(vEBT_vebt_member(Uu2),Uub)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).
% ATP.lambda_305
tff(fact_10514_ATP_Olambda__306,axiom,
! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_al(Uu2),Uua),Uub)
<=> ( aa_nat_o(vEBT_vebt_member(Uu2),Uub)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).
% ATP.lambda_306
tff(fact_10515_ATP_Olambda__307,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sh(Uu2),Uua),Uub)
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu2) ) ) ).
% ATP.lambda_307
tff(fact_10516_ATP_Olambda__308,axiom,
! [Uu2: set_complex,Uua: fun_complex_o,Uub: complex] :
( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_nw(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(member_complex(Uub),Uu2)
& aa_complex_o(Uua,Uub) ) ) ).
% ATP.lambda_308
tff(fact_10517_ATP_Olambda__309,axiom,
! [Uu2: set_real,Uua: fun_real_o,Uub: real] :
( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_nq(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& aa_real_o(Uua,Uub) ) ) ).
% ATP.lambda_309
tff(fact_10518_ATP_Olambda__310,axiom,
! [Uu2: set_nat,Uua: fun_nat_o,Uub: nat] :
( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ns(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& aa_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_310
tff(fact_10519_ATP_Olambda__311,axiom,
! [Uu2: set_int,Uua: fun_int_o,Uub: int] :
( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_nu(Uu2),Uua),Uub)
<=> ( aa_set_int_o(member_int(Uub),Uu2)
& aa_int_o(Uua,Uub) ) ) ).
% ATP.lambda_311
tff(fact_10520_ATP_Olambda__312,axiom,
! [Uu2: set_complex,Uua: fun_complex_complex,Uub: complex] :
( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_cq(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(member_complex(Uub),Uu2)
& ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_312
tff(fact_10521_ATP_Olambda__313,axiom,
! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_cy(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(member_complex(Uub),Uu2)
& ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_313
tff(fact_10522_ATP_Olambda__314,axiom,
! [Uu2: set_real,Uua: fun_real_complex,Uub: real] :
( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_ck(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_314
tff(fact_10523_ATP_Olambda__315,axiom,
! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_cs(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_315
tff(fact_10524_ATP_Olambda__316,axiom,
! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_da(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).
% ATP.lambda_316
tff(fact_10525_ATP_Olambda__317,axiom,
! [Uu2: set_nat,Uua: fun_nat_complex,Uub: nat] :
( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_cm(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_317
tff(fact_10526_ATP_Olambda__318,axiom,
! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_318
tff(fact_10527_ATP_Olambda__319,axiom,
! [Uu2: set_nat,Uua: fun_nat_rat,Uub: nat] :
( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_dc(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).
% ATP.lambda_319
tff(fact_10528_ATP_Olambda__320,axiom,
! [Uu2: set_int,Uua: fun_int_complex,Uub: int] :
( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_co(Uu2),Uua),Uub)
<=> ( aa_set_int_o(member_int(Uub),Uu2)
& ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_320
tff(fact_10529_ATP_Olambda__321,axiom,
! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_cw(Uu2),Uua),Uub)
<=> ( aa_set_int_o(member_int(Uub),Uu2)
& ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_321
tff(fact_10530_ATP_Olambda__322,axiom,
! [Uu2: set_complex,Uua: fun_complex_complex,Uub: complex] :
( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_bw(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(member_complex(Uub),Uu2)
& ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_322
tff(fact_10531_ATP_Olambda__323,axiom,
! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ce(Uu2),Uua),Uub)
<=> ( aa_set_complex_o(member_complex(Uub),Uu2)
& ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_323
tff(fact_10532_ATP_Olambda__324,axiom,
! [Uu2: set_real,Uua: fun_real_complex,Uub: real] :
( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_bq(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_324
tff(fact_10533_ATP_Olambda__325,axiom,
! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_by(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_325
tff(fact_10534_ATP_Olambda__326,axiom,
! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_cg(Uu2),Uua),Uub)
<=> ( aa_set_real_o(member_real(Uub),Uu2)
& ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).
% ATP.lambda_326
tff(fact_10535_ATP_Olambda__327,axiom,
! [Uu2: set_nat,Uua: fun_nat_complex,Uub: nat] :
( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_bs(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_327
tff(fact_10536_ATP_Olambda__328,axiom,
! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ca(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_328
tff(fact_10537_ATP_Olambda__329,axiom,
! [Uu2: set_nat,Uua: fun_nat_rat,Uub: nat] :
( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ci(Uu2),Uua),Uub)
<=> ( aa_set_nat_o(member_nat(Uub),Uu2)
& ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).
% ATP.lambda_329
tff(fact_10538_ATP_Olambda__330,axiom,
! [Uu2: set_int,Uua: fun_int_complex,Uub: int] :
( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_bu(Uu2),Uua),Uub)
<=> ( aa_set_int_o(member_int(Uub),Uu2)
& ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_330
tff(fact_10539_ATP_Olambda__331,axiom,
! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_cc(Uu2),Uua),Uub)
<=> ( aa_set_int_o(member_int(Uub),Uu2)
& ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_331
tff(fact_10540_ATP_Olambda__332,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rb(Uu2),Uua),Uub)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu2) ) ).
% ATP.lambda_332
tff(fact_10541_ATP_Olambda__333,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qm(Uu2),Uua),Uub)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu2) ) ).
% ATP.lambda_333
tff(fact_10542_ATP_Olambda__334,axiom,
! [Uu2: real,Uua: complex,Uub: complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_vn(Uu2),Uua),Uub)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu2) ) ).
% ATP.lambda_334
tff(fact_10543_ATP_Olambda__335,axiom,
! [Uu2: real,Uua: real,Uub: real] :
( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vl(Uu2),Uua),Uub)
<=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu2) ) ).
% ATP.lambda_335
tff(fact_10544_ATP_Olambda__336,axiom,
! [Uu2: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_ek(Uu2),Uua),(Uub))
<=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu2) ) ).
% ATP.lambda_336
tff(fact_10545_ATP_Olambda__337,axiom,
! [Uu2: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
( aa_num_o(aa_num_fun_num_o(aTP_Lamm_el(Uu2),Uua),Uub)
<=> member7279096912039735102um_num(product_Pair_num_num(Uua,Uub),Uu2) ) ).
% ATP.lambda_337
tff(fact_10546_ATP_Olambda__338,axiom,
! [Uu2: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_em(Uu2),Uua),Uub)
<=> member9148766508732265716at_num(product_Pair_nat_num(Uua,Uub),Uu2) ) ).
% ATP.lambda_338
tff(fact_10547_ATP_Olambda__339,axiom,
! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_en(Uu2),Uua),Uub)
<=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu2) ) ).
% ATP.lambda_339
tff(fact_10548_ATP_Olambda__340,axiom,
! [Uu2: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_eo(Uu2),Uua),Uub)
<=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu2) ) ).
% ATP.lambda_340
tff(fact_10549_ATP_Olambda__341,axiom,
! [Uu2: nat,Uua: complex,Uub: complex] :
( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_bn(Uu2),Uua),Uub)
<=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).
% ATP.lambda_341
tff(fact_10550_ATP_Olambda__342,axiom,
! [Uu2: complex,Uua: nat,Uub: complex] :
( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_rs(Uu2),Uua),Uub)
<=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).
% ATP.lambda_342
tff(fact_10551_ATP_Olambda__343,axiom,
! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_iz(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_343
tff(fact_10552_ATP_Olambda__344,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ja(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_344
tff(fact_10553_ATP_Olambda__345,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_ul(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).
% ATP.lambda_345
tff(fact_10554_ATP_Olambda__346,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tv(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_346
tff(fact_10555_ATP_Olambda__347,axiom,
! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_iw(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_347
tff(fact_10556_ATP_Olambda__348,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iu(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_348
tff(fact_10557_ATP_Olambda__349,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qu(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).
% ATP.lambda_349
tff(fact_10558_ATP_Olambda__350,axiom,
! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aw(Uu2),Uua),Uub)
<=> ( aa_nat_o(Uu2,Uub)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).
% ATP.lambda_350
tff(fact_10559_ATP_Olambda__351,axiom,
! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vj(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).
% ATP.lambda_351
tff(fact_10560_ATP_Olambda__352,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nf(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_352
tff(fact_10561_ATP_Olambda__353,axiom,
! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ng(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).
% ATP.lambda_353
tff(fact_10562_ATP_Olambda__354,axiom,
! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_nh(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).
% ATP.lambda_354
tff(fact_10563_ATP_Olambda__355,axiom,
! [Uu2: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_fl(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu2,Uub)) ).
% ATP.lambda_355
tff(fact_10564_ATP_Olambda__356,axiom,
! [Uu2: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fp(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu2,Uub)) ).
% ATP.lambda_356
tff(fact_10565_ATP_Olambda__357,axiom,
! [Uu2: fun_complex_int,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_ft(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(Uua,Uub)),aa_complex_int(Uu2,Uub)) ).
% ATP.lambda_357
tff(fact_10566_ATP_Olambda__358,axiom,
! [Uu2: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_fm(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu2,Uub)) ).
% ATP.lambda_358
tff(fact_10567_ATP_Olambda__359,axiom,
! [Uu2: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fq(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu2,Uub)) ).
% ATP.lambda_359
tff(fact_10568_ATP_Olambda__360,axiom,
! [Uu2: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_fu(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu2,Uub)) ).
% ATP.lambda_360
tff(fact_10569_ATP_Olambda__361,axiom,
! [Uu2: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_fn(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu2,Uub)) ).
% ATP.lambda_361
tff(fact_10570_ATP_Olambda__362,axiom,
! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fr(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu2,Uub)) ).
% ATP.lambda_362
tff(fact_10571_ATP_Olambda__363,axiom,
! [Uu2: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_fo(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu2,Uub)) ).
% ATP.lambda_363
tff(fact_10572_ATP_Olambda__364,axiom,
! [Uu2: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fs(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu2,Uub)) ).
% ATP.lambda_364
tff(fact_10573_ATP_Olambda__365,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ub(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_365
tff(fact_10574_ATP_Olambda__366,axiom,
! [Uu2: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_gb(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu2,Uub)) ).
% ATP.lambda_366
tff(fact_10575_ATP_Olambda__367,axiom,
! [Uu2: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gc(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu2,Uub)) ).
% ATP.lambda_367
tff(fact_10576_ATP_Olambda__368,axiom,
! [Uu2: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gd(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu2,Uub)) ).
% ATP.lambda_368
tff(fact_10577_ATP_Olambda__369,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lg(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu2,Uub)) ).
% ATP.lambda_369
tff(fact_10578_ATP_Olambda__370,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gf(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu2,Uub)) ).
% ATP.lambda_370
tff(fact_10579_ATP_Olambda__371,axiom,
! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ge(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu2,Uub)) ).
% ATP.lambda_371
tff(fact_10580_ATP_Olambda__372,axiom,
! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fi(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uub)) ).
% ATP.lambda_372
tff(fact_10581_ATP_Olambda__373,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fk(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_373
tff(fact_10582_ATP_Olambda__374,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fj(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_374
tff(fact_10583_ATP_Olambda__375,axiom,
! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ip(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).
% ATP.lambda_375
tff(fact_10584_ATP_Olambda__376,axiom,
! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fh(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).
% ATP.lambda_376
tff(fact_10585_ATP_Olambda__377,axiom,
! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tl(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).
% ATP.lambda_377
tff(fact_10586_ATP_Olambda__378,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ax(Uu2),Uua),Uub)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua) ) ).
% ATP.lambda_378
tff(fact_10587_ATP_Olambda__379,axiom,
! [Uu2: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ew(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu2,Uub)),Uua) ).
% ATP.lambda_379
tff(fact_10588_ATP_Olambda__380,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ey(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),Uua) ).
% ATP.lambda_380
tff(fact_10589_ATP_Olambda__381,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ex(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),Uua) ).
% ATP.lambda_381
tff(fact_10590_ATP_Olambda__382,axiom,
! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ev(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),Uua) ).
% ATP.lambda_382
tff(fact_10591_ATP_Olambda__383,axiom,
! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ky(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu2) ).
% ATP.lambda_383
tff(fact_10592_ATP_Olambda__384,axiom,
! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kz(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu2) ).
% ATP.lambda_384
tff(fact_10593_ATP_Olambda__385,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jl(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua) ).
% ATP.lambda_385
tff(fact_10594_ATP_Olambda__386,axiom,
! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_jj(Uu2),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uub)),Uua) ).
% ATP.lambda_386
tff(fact_10595_ATP_Olambda__387,axiom,
! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_jk(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uub)),Uua) ).
% ATP.lambda_387
tff(fact_10596_ATP_Olambda__388,axiom,
! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ti(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu2,Uub)),Uua) ).
% ATP.lambda_388
tff(fact_10597_ATP_Olambda__389,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ni(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu2,Uub)),Uua) ).
% ATP.lambda_389
tff(fact_10598_ATP_Olambda__390,axiom,
! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nj(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu2,Uub)),Uua) ).
% ATP.lambda_390
tff(fact_10599_ATP_Olambda__391,axiom,
! [Uu2: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_nk(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu2,Uub)),Uua) ).
% ATP.lambda_391
tff(fact_10600_ATP_Olambda__392,axiom,
! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_uq(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ).
% ATP.lambda_392
tff(fact_10601_ATP_Olambda__393,axiom,
! [Uu2: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tk(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),Uua) ).
% ATP.lambda_393
tff(fact_10602_ATP_Olambda__394,axiom,
! [Uu2: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_jn(Uu2),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,Uu2),Uub)) ).
% ATP.lambda_394
tff(fact_10603_ATP_Olambda__395,axiom,
! [Uu2: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_hs(Uu2),Uua),Uub) = plus_p3455044024723400733d_enat(Uu2,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).
% ATP.lambda_395
tff(fact_10604_ATP_Olambda__396,axiom,
! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ht(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).
% ATP.lambda_396
tff(fact_10605_ATP_Olambda__397,axiom,
! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).
% ATP.lambda_397
tff(fact_10606_ATP_Olambda__398,axiom,
! [Uu2: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_hr(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu2),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).
% ATP.lambda_398
tff(fact_10607_ATP_Olambda__399,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hv(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).
% ATP.lambda_399
tff(fact_10608_ATP_Olambda__400,axiom,
! [Uu2: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hu(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).
% ATP.lambda_400
tff(fact_10609_ATP_Olambda__401,axiom,
! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ke(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_401
tff(fact_10610_ATP_Olambda__402,axiom,
! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_kh(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_402
tff(fact_10611_ATP_Olambda__403,axiom,
! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_kf(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_403
tff(fact_10612_ATP_Olambda__404,axiom,
! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_kg(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_404
tff(fact_10613_ATP_Olambda__405,axiom,
! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ug(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).
% ATP.lambda_405
tff(fact_10614_ATP_Olambda__406,axiom,
! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_gk(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_406
tff(fact_10615_ATP_Olambda__407,axiom,
! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gn(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_407
tff(fact_10616_ATP_Olambda__408,axiom,
! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_gl(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_408
tff(fact_10617_ATP_Olambda__409,axiom,
! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_gm(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_409
tff(fact_10618_ATP_Olambda__410,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hg(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).
% ATP.lambda_410
tff(fact_10619_ATP_Olambda__411,axiom,
! [Uu2: real,Uua: real,Uub: real] :
( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ut(Uu2),Uua),Uub)
<=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu2,Uua)) ) ).
% ATP.lambda_411
tff(fact_10620_ATP_Olambda__412,axiom,
! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_es(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(Uu2),aa_complex_complex(Uua,Uub)) ).
% ATP.lambda_412
tff(fact_10621_ATP_Olambda__413,axiom,
! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_in(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(Uu2),aa_nat_complex(Uua,Uub)) ).
% ATP.lambda_413
tff(fact_10622_ATP_Olambda__414,axiom,
! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_eu(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_414
tff(fact_10623_ATP_Olambda__415,axiom,
! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_et(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_415
tff(fact_10624_ATP_Olambda__416,axiom,
! [Uu2: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_er(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),aa_int_int(Uua,Uub)) ).
% ATP.lambda_416
tff(fact_10625_ATP_Olambda__417,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_io(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu2,Uub)) ).
% ATP.lambda_417
tff(fact_10626_ATP_Olambda__418,axiom,
! [Uu2: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_of(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_418
tff(fact_10627_ATP_Olambda__419,axiom,
! [Uu2: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_oe(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_419
tff(fact_10628_ATP_Olambda__420,axiom,
! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_og(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_420
tff(fact_10629_ATP_Olambda__421,axiom,
! [Uu2: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_oh(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_421
tff(fact_10630_ATP_Olambda__422,axiom,
! [Uu2: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_oi(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_int_nat(Uua,Uub)) ).
% ATP.lambda_422
tff(fact_10631_ATP_Olambda__423,axiom,
! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).
% ATP.lambda_423
tff(fact_10632_ATP_Olambda__424,axiom,
! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_pm(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).
% ATP.lambda_424
tff(fact_10633_ATP_Olambda__425,axiom,
! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_po(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).
% ATP.lambda_425
tff(fact_10634_ATP_Olambda__426,axiom,
! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pp(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).
% ATP.lambda_426
tff(fact_10635_ATP_Olambda__427,axiom,
! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jf(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_427
tff(fact_10636_ATP_Olambda__428,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_je(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_428
tff(fact_10637_ATP_Olambda__429,axiom,
! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ou(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_429
tff(fact_10638_ATP_Olambda__430,axiom,
! [Uu2: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_jm(Uu2),Uua),Uub) = aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_430
tff(fact_10639_ATP_Olambda__431,axiom,
! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gj(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_431
tff(fact_10640_ATP_Olambda__432,axiom,
! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_us(Uu2),Uua),Uub)
<=> aa_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).
% ATP.lambda_432
tff(fact_10641_ATP_Olambda__433,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gi(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_433
tff(fact_10642_ATP_Olambda__434,axiom,
! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oj(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_434
tff(fact_10643_ATP_Olambda__435,axiom,
! [Uu2: fun_Pr2602117396471556196at_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ds(Uu2),Uua),Uub) = aa_Pro4243173770044397973at_nat(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).
% ATP.lambda_435
tff(fact_10644_ATP_Olambda__436,axiom,
! [Uu2: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_dt(Uu2),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).
% ATP.lambda_436
tff(fact_10645_ATP_Olambda__437,axiom,
! [Uu2: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_du(Uu2),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).
% ATP.lambda_437
tff(fact_10646_ATP_Olambda__438,axiom,
! [Uu2: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dv(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).
% ATP.lambda_438
tff(fact_10647_ATP_Olambda__439,axiom,
! [Uu2: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dw(Uu2),Uua),Uub)
<=> aa_Pro4592068669816069595_int_o(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).
% ATP.lambda_439
tff(fact_10648_ATP_Olambda__440,axiom,
! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lb(Uu2),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu2)) ).
% ATP.lambda_440
tff(fact_10649_ATP_Olambda__441,axiom,
! [Uu2: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lc(Uu2),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu2)) ).
% ATP.lambda_441
tff(fact_10650_ATP_Olambda__442,axiom,
! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_ix(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).
% ATP.lambda_442
tff(fact_10651_ATP_Olambda__443,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iv(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).
% ATP.lambda_443
tff(fact_10652_ATP_Olambda__444,axiom,
! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uk(Uu2),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua)) ).
% ATP.lambda_444
tff(fact_10653_ATP_Olambda__445,axiom,
! [Uu2: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_lh(Uu2),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu2,Uuc)) ).
% ATP.lambda_445
tff(fact_10654_ATP_Olambda__446,axiom,
! [Uu2: 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_hh(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_real(Uua,Uuc),aa_VEBT_VEBT_real(Uub,Uuc)) ).
% ATP.lambda_446
tff(fact_10655_ATP_Olambda__447,axiom,
! [Uu2: 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_hj(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ).
% ATP.lambda_447
tff(fact_10656_ATP_Olambda__448,axiom,
! [Uu2: 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_hl(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_nat(Uua,Uuc),aa_VEBT_VEBT_nat(Uub,Uuc)) ).
% ATP.lambda_448
tff(fact_10657_ATP_Olambda__449,axiom,
! [Uu2: 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_hn(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_int(Uua,Uuc),aa_VEBT_VEBT_int(Uub,Uuc)) ).
% ATP.lambda_449
tff(fact_10658_ATP_Olambda__450,axiom,
! [Uu2: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_hi(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).
% ATP.lambda_450
tff(fact_10659_ATP_Olambda__451,axiom,
! [Uu2: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_hk(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).
% ATP.lambda_451
tff(fact_10660_ATP_Olambda__452,axiom,
! [Uu2: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_hm(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).
% ATP.lambda_452
tff(fact_10661_ATP_Olambda__453,axiom,
! [Uu2: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_ho(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).
% ATP.lambda_453
tff(fact_10662_ATP_Olambda__454,axiom,
! [Uu2: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_pb(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ).
% ATP.lambda_454
tff(fact_10663_ATP_Olambda__455,axiom,
! [Uu2: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_pd(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) ).
% ATP.lambda_455
tff(fact_10664_ATP_Olambda__456,axiom,
! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_pc(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).
% ATP.lambda_456
tff(fact_10665_ATP_Olambda__457,axiom,
! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_pe(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).
% ATP.lambda_457
tff(fact_10666_ATP_Olambda__458,axiom,
! [Uu2: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_hp(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_int_real(Uua,Uuc),aa_int_real(Uub,Uuc)) ).
% ATP.lambda_458
tff(fact_10667_ATP_Olambda__459,axiom,
! [Uu2: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_hq(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).
% ATP.lambda_459
tff(fact_10668_ATP_Olambda__460,axiom,
! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_im(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_il(Uu2),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_460
tff(fact_10669_ATP_Olambda__461,axiom,
! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ik(Uu2),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ij(Uu2),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_461
tff(fact_10670_ATP_Olambda__462,axiom,
! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ii(Uu2),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ih(Uu2),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_462
tff(fact_10671_ATP_Olambda__463,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_fg(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ff(Uu2),Uua),Uuc),Uub) ).
% ATP.lambda_463
tff(fact_10672_ATP_Olambda__464,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_fe(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_fd(Uu2),Uua),Uuc)),Uub) ).
% ATP.lambda_464
tff(fact_10673_ATP_Olambda__465,axiom,
! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_fc(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_fb(Uu2),Uua),Uuc),Uub) ).
% ATP.lambda_465
tff(fact_10674_ATP_Olambda__466,axiom,
! [Uu2: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_fa(Uu2),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ez(Uu2),Uua),Uuc),Uub) ).
% ATP.lambda_466
tff(fact_10675_ATP_Olambda__467,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qw(Uu2),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_qv(Uu2),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_467
tff(fact_10676_ATP_Olambda__468,axiom,
! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tr(Uu2),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(Uu2,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_468
tff(fact_10677_ATP_Olambda__469,axiom,
! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tp(Uu2),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(Uu2,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_469
tff(fact_10678_ATP_Olambda__470,axiom,
! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_tn(Uu2),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(Uu2,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_470
tff(fact_10679_ATP_Olambda__471,axiom,
! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_to(Uu2),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(Uu2,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_471
tff(fact_10680_ATP_Olambda__472,axiom,
! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jo(Uu2),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(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uub)) ).
% ATP.lambda_472
tff(fact_10681_ATP_Olambda__473,axiom,
! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ju(Uu2),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,Uu2),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,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).
% ATP.lambda_473
tff(fact_10682_ATP_Olambda__474,axiom,
! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jq(Uu2),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uub)) ).
% ATP.lambda_474
tff(fact_10683_ATP_Olambda__475,axiom,
! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_js(Uu2),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,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uu2),Uub)) ).
% ATP.lambda_475
tff(fact_10684_ATP_Olambda__476,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_si(Uu2),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,Uu2),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,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).
% ATP.lambda_476
tff(fact_10685_ATP_Olambda__477,axiom,
! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ih(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu2)),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_477
tff(fact_10686_ATP_Olambda__478,axiom,
! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_il(Uu2),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),Uu2)),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_478
tff(fact_10687_ATP_Olambda__479,axiom,
! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ij(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu2)),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_479
tff(fact_10688_ATP_Olambda__480,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qr(Uu2),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,Uu2),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_480
tff(fact_10689_ATP_Olambda__481,axiom,
! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ka(Uu2),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(Uu2),Uuc)) ).
% ATP.lambda_481
tff(fact_10690_ATP_Olambda__482,axiom,
! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_kd(Uu2),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,Uu2),Uuc)) ).
% ATP.lambda_482
tff(fact_10691_ATP_Olambda__483,axiom,
! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_kb(Uu2),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(Uu2),Uuc)) ).
% ATP.lambda_483
tff(fact_10692_ATP_Olambda__484,axiom,
! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_kc(Uu2),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(Uu2),Uuc)) ).
% ATP.lambda_484
tff(fact_10693_ATP_Olambda__485,axiom,
! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jp(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_485
tff(fact_10694_ATP_Olambda__486,axiom,
! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jv(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),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,Uu2),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,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_486
tff(fact_10695_ATP_Olambda__487,axiom,
! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jr(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_487
tff(fact_10696_ATP_Olambda__488,axiom,
! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_jt(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),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,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_488
tff(fact_10697_ATP_Olambda__489,axiom,
! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jw(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_489
tff(fact_10698_ATP_Olambda__490,axiom,
! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jz(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),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_490
tff(fact_10699_ATP_Olambda__491,axiom,
! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_jx(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_491
tff(fact_10700_ATP_Olambda__492,axiom,
! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jy(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_492
tff(fact_10701_ATP_Olambda__493,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ql(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),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_493
tff(fact_10702_ATP_Olambda__494,axiom,
! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_rg(Uu2),Uua),Uub),Uuc)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).
% ATP.lambda_494
tff(fact_10703_ATP_Olambda__495,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sn(Uu2),Uua),Uub),Uuc)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).
% ATP.lambda_495
tff(fact_10704_ATP_Olambda__496,axiom,
! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_re(Uu2),Uua),Uub),Uuc)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).
% ATP.lambda_496
tff(fact_10705_ATP_Olambda__497,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sl(Uu2),Uua),Uub),Uuc)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).
% ATP.lambda_497
tff(fact_10706_ATP_Olambda__498,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sp(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).
% ATP.lambda_498
tff(fact_10707_ATP_Olambda__499,axiom,
! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sr(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_499
tff(fact_10708_ATP_Olambda__500,axiom,
! [Uu2: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_bx(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
& ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_500
tff(fact_10709_ATP_Olambda__501,axiom,
! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_cf(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_501
tff(fact_10710_ATP_Olambda__502,axiom,
! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_br(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_real_o(member_real(Uuc),Uu2)
& ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_502
tff(fact_10711_ATP_Olambda__503,axiom,
! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_bz(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_real_o(member_real(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_503
tff(fact_10712_ATP_Olambda__504,axiom,
! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ch(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_real_o(member_real(Uuc),Uu2)
& ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).
% ATP.lambda_504
tff(fact_10713_ATP_Olambda__505,axiom,
! [Uu2: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_bt(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
& ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_505
tff(fact_10714_ATP_Olambda__506,axiom,
! [Uu2: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_cb(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_506
tff(fact_10715_ATP_Olambda__507,axiom,
! [Uu2: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_cj(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
& ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).
% ATP.lambda_507
tff(fact_10716_ATP_Olambda__508,axiom,
! [Uu2: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_bv(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_int_o(member_int(Uuc),Uu2)
& ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_508
tff(fact_10717_ATP_Olambda__509,axiom,
! [Uu2: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_cd(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_int_o(member_int(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_509
tff(fact_10718_ATP_Olambda__510,axiom,
! [Uu2: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_cr(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
& ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_510
tff(fact_10719_ATP_Olambda__511,axiom,
! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_cz(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_511
tff(fact_10720_ATP_Olambda__512,axiom,
! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_cl(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_real_o(member_real(Uuc),Uu2)
& ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_512
tff(fact_10721_ATP_Olambda__513,axiom,
! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ct(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_real_o(member_real(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_513
tff(fact_10722_ATP_Olambda__514,axiom,
! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_db(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_real_o(member_real(Uuc),Uu2)
& ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).
% ATP.lambda_514
tff(fact_10723_ATP_Olambda__515,axiom,
! [Uu2: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_cn(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
& ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_515
tff(fact_10724_ATP_Olambda__516,axiom,
! [Uu2: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_cv(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_516
tff(fact_10725_ATP_Olambda__517,axiom,
! [Uu2: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dd(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
& ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).
% ATP.lambda_517
tff(fact_10726_ATP_Olambda__518,axiom,
! [Uu2: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_cp(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_int_o(member_int(Uuc),Uu2)
& ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_518
tff(fact_10727_ATP_Olambda__519,axiom,
! [Uu2: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_cx(Uu2),Uua),Uub),Uuc)
<=> ( aa_set_int_o(member_int(Uuc),Uu2)
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_519
tff(fact_10728_ATP_Olambda__520,axiom,
! [Uu2: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jb(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).
% ATP.lambda_520
tff(fact_10729_ATP_Olambda__521,axiom,
! [Uu2: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jc(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,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_521
tff(fact_10730_ATP_Olambda__522,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qv(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).
% ATP.lambda_522
tff(fact_10731_ATP_Olambda__523,axiom,
! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_fb(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uuc)) ).
% ATP.lambda_523
tff(fact_10732_ATP_Olambda__524,axiom,
! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ff(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uuc)) ).
% ATP.lambda_524
tff(fact_10733_ATP_Olambda__525,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_fd(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uuc)) ).
% ATP.lambda_525
tff(fact_10734_ATP_Olambda__526,axiom,
! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ez(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uuc)) ).
% ATP.lambda_526
tff(fact_10735_ATP_Olambda__527,axiom,
! [Uu2: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_eg((Uu2)),Uua),Uub),Uuc)
<=> ( (Uu2)
& aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).
% ATP.lambda_527
tff(fact_10736_ATP_Olambda__528,axiom,
! [Uu2: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_do(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).
% ATP.lambda_528
tff(fact_10737_ATP_Olambda__529,axiom,
! [Uu2: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_dk(Uu2),Uua),Uub),Uuc)
<=> aa_Pro4592068669816069595_int_o(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).
% ATP.lambda_529
tff(fact_10738_ATP_Olambda__530,axiom,
! [Uu2: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_dl(Uu2),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu2,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).
% ATP.lambda_530
tff(fact_10739_ATP_Olambda__531,axiom,
! [Uu2: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_dj(Uu2),Uua),Uub),Uuc)
<=> aa_o_o(Uu2,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).
% ATP.lambda_531
tff(fact_10740_ATP_Olambda__532,axiom,
! [Uu2: fun_nat_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_di(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).
% ATP.lambda_532
tff(fact_10741_ATP_Olambda__533,axiom,
! [Uu2: fun_na8713714802964096592at_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aa_fun2484538639213223237at_nat(aTP_Lamm_dp(Uu2),Uua),Uub),Uuc) = aa_nat7997280740862404369at_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).
% ATP.lambda_533
tff(fact_10742_ATP_Olambda__534,axiom,
! [Uu2: fun_na3970208099681795815_nat_o,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun2100511394839469148_nat_o(aTP_Lamm_dm(Uu2),Uua),Uub),Uuc) = aa_nat7695548938202568488_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).
% ATP.lambda_534
tff(fact_10743_ATP_Olambda__535,axiom,
! [Uu2: fun_fu1413608398870722896at_nat,Uua: fun_na1668100129766719005at_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun8817627472026143685at_nat(aTP_Lamm_dq(Uu2),Uua),Uub),Uuc) = aa_fun7582795104022253201at_nat(Uu2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uua,Uub),Uuc)) ).
% ATP.lambda_535
tff(fact_10744_ATP_Olambda__536,axiom,
! [Uu2: fun_fu1115759055147092071_o_nat,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun3680403850923830236at_nat(aTP_Lamm_dn(Uu2),Uua),Uub),Uuc) = aa_fun1432809138467554728_o_nat(Uu2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).
% ATP.lambda_536
tff(fact_10745_ATP_Olambda__537,axiom,
! [Uu2: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_dr(Uu2),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).
% ATP.lambda_537
tff(fact_10746_ATP_Olambda__538,axiom,
! [Uu2: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gp(Uu2),Uua),Uub),Uuc) = aa_nat_real(Uu2,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_538
tff(fact_10747_ATP_Olambda__539,axiom,
! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_go(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,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_539
tff(fact_10748_ATP_Olambda__540,axiom,
! [Uu2: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ov(Uu2),Uua),Uub),Uuc) = aa_nat_int(Uu2,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_540
tff(fact_10749_ATP_Olambda__541,axiom,
! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rm(Uu2),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,Uu2),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_541
tff(fact_10750_ATP_Olambda__542,axiom,
! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rj(Uu2),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(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),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_542
tff(fact_10751_ATP_Olambda__543,axiom,
! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rq(Uu2),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,Uu2),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_543
tff(fact_10752_ATP_Olambda__544,axiom,
! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ro(Uu2),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,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).
% ATP.lambda_544
tff(fact_10753_ATP_Olambda__545,axiom,
! [Uu2: 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_tq(Uu2),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_tp(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),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,Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))))) ).
% ATP.lambda_545
tff(fact_10754_ATP_Olambda__546,axiom,
! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_uu(Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
& ! [I4: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Uud))
=> ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I4)),X_12)
<=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I4) ) )
& $ite(
Uue = Uuf,
! [X: vEBT_VEBT] :
( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Uua))
=> ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X9) ),
( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
& ! [X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(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))),Uu2))
=> ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Uuf) ) ) ) ) ) ) ) ).
% ATP.lambda_546
tff(fact_10755_ATP_Olambda__547,axiom,
! [Uu2: int,Uua: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_eh,Uu2),Uua)
<=> $true ) ).
% ATP.lambda_547
tff(fact_10756_ATP_Olambda__548,axiom,
! [Uu2: complex] : aa_complex_complex(aTP_Lamm_ag,Uu2) = Uu2 ).
% ATP.lambda_548
tff(fact_10757_ATP_Olambda__549,axiom,
! [Uu2: real] : aa_real_real(aTP_Lamm_ah,Uu2) = Uu2 ).
% ATP.lambda_549
tff(fact_10758_ATP_Olambda__550,axiom,
! [Uu2: rat] : aa_rat_rat(aTP_Lamm_ai,Uu2) = Uu2 ).
% ATP.lambda_550
tff(fact_10759_ATP_Olambda__551,axiom,
! [Uu2: nat] : aa_nat_nat(aTP_Lamm_aj,Uu2) = Uu2 ).
% ATP.lambda_551
tff(fact_10760_ATP_Olambda__552,axiom,
! [Uu2: int] : aa_int_int(aTP_Lamm_ak,Uu2) = Uu2 ).
% ATP.lambda_552
tff(fact_10761_ATP_Olambda__553,axiom,
! [Uu2: complex] : aa_complex_complex(aTP_Lamm_an,Uu2) = zero_zero_complex ).
% ATP.lambda_553
tff(fact_10762_ATP_Olambda__554,axiom,
! [Uu2: real] : aa_real_real(aTP_Lamm_ao,Uu2) = zero_zero_real ).
% ATP.lambda_554
tff(fact_10763_ATP_Olambda__555,axiom,
! [Uu2: rat] : aa_rat_rat(aTP_Lamm_ap,Uu2) = zero_zero_rat ).
% ATP.lambda_555
tff(fact_10764_ATP_Olambda__556,axiom,
! [Uu2: nat] : aa_nat_nat(aTP_Lamm_aq,Uu2) = zero_zero_nat ).
% ATP.lambda_556
tff(fact_10765_ATP_Olambda__557,axiom,
! [Uu2: int] : aa_int_int(aTP_Lamm_ar,Uu2) = zero_zero_int ).
% ATP.lambda_557
tff(fact_10766_ATP_Olambda__558,axiom,
! [Uu2: nat] : aa_nat_nat(aTP_Lamm_mi,Uu2) = one_one_nat ).
% ATP.lambda_558
tff(fact_10767_ATP_Olambda__559,axiom,
! [Uu2: nat] : aa_nat_int(aTP_Lamm_mj,Uu2) = one_one_int ).
% ATP.lambda_559
tff(fact_10768_ATP_Olambda__560,axiom,
! [Uu2: int] : aa_int_int(aTP_Lamm_mk,Uu2) = one_one_int ).
% ATP.lambda_560
tff(fact_10769_ATP_Olambda__561,axiom,
! [Uu2: nat] :
( aa_nat_o(aTP_Lamm_ry,Uu2)
<=> $false ) ).
% ATP.lambda_561
tff(fact_10770_ATP_Olambda__562,axiom,
! [Uu2: nat] :
( aa_nat_o(aTP_Lamm_rx,Uu2)
<=> $true ) ).
% ATP.lambda_562
% Helper facts (1)
tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P: fun_real_o] :
( aa_real_o(P,fChoice_real(P))
= ( ? [X9: real] : aa_real_o(P,X9) ) ) ).
% Conjectures (1)
tff(conj_0,conjecture,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(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,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)))) ).
%------------------------------------------------------------------------------