TPTP Problem File: ITP217_1.p
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%------------------------------------------------------------------------------
% File : ITP217_1 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Time_Reasoning 00330_012302
% 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 : 0029_Time_Reasoning_00330_012302 [Des22]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.70 v8.1.0
% Syntax : Number of formulae : 7749 (2513 unt;1449 typ; 0 def)
% Number of atoms : 14413 (5179 equ)
% Maximal formula atoms : 38 ( 2 avg)
% Number of connectives : 9509 (1396 ~; 327 |; 809 &)
% (1589 <=>;5388 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 4 avg)
% Maximal term depth : 40 ( 2 avg)
% Number of types : 295 ( 294 usr)
% Number of type conns : 1424 ( 788 >; 636 *; 0 +; 0 <<)
% Number of predicates : 82 ( 79 usr; 2 prp; 0-4 aty)
% Number of functors : 1076 (1076 usr; 367 con; 0-7 aty)
% Number of variables : 13880 (13616 !; 264 ?;13880 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 16:31:48.940
%------------------------------------------------------------------------------
% Could-be-implicit typings (294)
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aTP_Lamm_au: set_se6260736226359567993nt_int > fun_fu6908345394967445887t_bool ).
tff(sy_c_ATP_058Lamm__av____,type,
aTP_Lamm_av: set_set_nat > fun_fu7495562154832025027t_bool ).
tff(sy_c_ATP_058Lamm__aw____,type,
aTP_Lamm_aw: set_nat > fun_fu3889889255038636099t_bool ).
tff(sy_c_ATP_058Lamm__ax____,type,
aTP_Lamm_ax: set_int > fun_fu7822069302422884163t_bool ).
tff(sy_c_ATP_058Lamm__ay____,type,
aTP_Lamm_ay: product_unit > fun_fu3400720548862938179Heap_a ).
tff(sy_c_ATP_058Lamm__az____,type,
aTP_Lamm_az: product_unit > fun_fu7990100517307520643t_unit ).
tff(sy_c_ATP_058Lamm__ba____,type,
aTP_Lamm_ba: a > fun_fu4828475307307880594Heap_a ).
tff(sy_c_ATP_058Lamm__bb____,type,
aTP_Lamm_bb: fun_Product_unit_a > fun_Pr5215307506330550056Heap_a ).
tff(sy_c_ATP_058Lamm__bc____,type,
aTP_Lamm_bc: fun_Pr2191293291710260995t_unit > fun_Pr292408699144966967t_unit ).
tff(sy_c_ATP_058Lamm__bd____,type,
aTP_Lamm_bd: fun_a_a > fun_a_6093291656547882009Heap_a ).
tff(sy_c_ATP_058Lamm__be____,type,
aTP_Lamm_be: fun_Pr2191293291710260995t_unit > fun_Pr5417461343077184813t_unit ).
tff(sy_c_ATP_058Lamm__bf____,type,
aTP_Lamm_bf: fun_Pr2191293291710260995t_unit > fun_re5565935564814528228t_unit ).
tff(sy_c_ATP_058Lamm__bg____,type,
aTP_Lamm_bg: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__bh____,type,
aTP_Lamm_bh: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__bi____,type,
aTP_Lamm_bi: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bj____,type,
aTP_Lamm_bj: fun_int_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bk____,type,
aTP_Lamm_bk: fun_bool_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bl____,type,
aTP_Lamm_bl: fun_Co7398788559141407565atural ).
tff(sy_c_ATP_058Lamm__bm____,type,
aTP_Lamm_bm: fun_Co8499238673658658103nteger ).
tff(sy_c_ATP_058Lamm__bn____,type,
aTP_Lamm_bn: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__bo____,type,
aTP_Lamm_bo: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bp____,type,
aTP_Lamm_bp: fun_int_int ).
tff(sy_c_ATP_058Lamm__bq____,type,
aTP_Lamm_bq: fun_Co7398788559141407565atural ).
tff(sy_c_ATP_058Lamm__br____,type,
aTP_Lamm_br: fun_Co8499238673658658103nteger ).
tff(sy_c_ATP_058Lamm__bs____,type,
aTP_Lamm_bs: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__bt____,type,
aTP_Lamm_bt: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bu____,type,
aTP_Lamm_bu: fun_int_int ).
tff(sy_c_ATP_058Lamm__bv____,type,
aTP_Lamm_bv: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__bw____,type,
aTP_Lamm_bw: int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__bx____,type,
aTP_Lamm_bx: code_integer > fun_na7078387940644931716nteger ).
tff(sy_c_ATP_058Lamm__by____,type,
aTP_Lamm_by: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__bz____,type,
aTP_Lamm_bz: code_natural > fun_na607345901702507418atural ).
tff(sy_c_ATP_058Lamm__ca____,type,
aTP_Lamm_ca: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cb____,type,
aTP_Lamm_cb: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__cc____,type,
aTP_Lamm_cc: fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__cd____,type,
aTP_Lamm_cd: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__ce____,type,
aTP_Lamm_ce: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__cf____,type,
aTP_Lamm_cf: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cg____,type,
aTP_Lamm_cg: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ch____,type,
aTP_Lamm_ch: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ci____,type,
aTP_Lamm_ci: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cj____,type,
aTP_Lamm_cj: nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__ck____,type,
aTP_Lamm_ck: nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__cl____,type,
aTP_Lamm_cl: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cm____,type,
aTP_Lamm_cm: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cn____,type,
aTP_Lamm_cn: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__co____,type,
aTP_Lamm_co: fun_nat_bool > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__cp____,type,
aTP_Lamm_cp: fun_nat_nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__cq____,type,
aTP_Lamm_cq: fun_nat_bool ).
tff(sy_c_ATP_058Lamm__cr____,type,
aTP_Lamm_cr: fun_nat_bool ).
tff(sy_c_ATP_058Lamm__cs____,type,
aTP_Lamm_cs: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__ct____,type,
aTP_Lamm_ct: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__cu____,type,
aTP_Lamm_cu: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__cv____,type,
aTP_Lamm_cv: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__cw____,type,
aTP_Lamm_cw: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__cx____,type,
aTP_Lamm_cx: fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__cy____,type,
aTP_Lamm_cy: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cz____,type,
aTP_Lamm_cz: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__da____,type,
aTP_Lamm_da: set_nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__db____,type,
aTP_Lamm_db: set_nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__dc____,type,
aTP_Lamm_dc: num > fun_num_int ).
tff(sy_c_ATP_058Lamm__dd____,type,
aTP_Lamm_dd: fun_num_option_num ).
tff(sy_c_ATP_058Lamm__de____,type,
aTP_Lamm_de: fun_int_nat ).
tff(sy_c_ATP_058Lamm__df____,type,
aTP_Lamm_df: int > fun_int_int ).
tff(sy_c_ATP_058Lamm__dg____,type,
aTP_Lamm_dg: code_integer > fun_Co8499238673658658103nteger ).
tff(sy_c_ATP_058Lamm__dh____,type,
aTP_Lamm_dh: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__di____,type,
aTP_Lamm_di: fun_num_option_num ).
tff(sy_c_ATP_058Lamm__dj____,type,
aTP_Lamm_dj: fun_Co3583066429914935172er_nat ).
tff(sy_c_ATP_058Lamm__dk____,type,
aTP_Lamm_dk: fun_Co8521658056209057742er_num ).
tff(sy_c_ATP_058Lamm__dl____,type,
aTP_Lamm_dl: fun_Co1092983284983208416er_int ).
tff(sy_c_ATP_058Lamm__dm____,type,
aTP_Lamm_dm: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__dn____,type,
aTP_Lamm_dn: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__do____,type,
aTP_Lamm_do: fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__dp____,type,
aTP_Lamm_dp: num > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__dq____,type,
aTP_Lamm_dq: num > fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__dr____,type,
aTP_Lamm_dr: num > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__ds____,type,
aTP_Lamm_ds: nat > fun_num_option_num ).
tff(sy_c_ATP_058Lamm__dt____,type,
aTP_Lamm_dt: nat > fun_num_option_num ).
tff(sy_c_ATP_058Lamm__du____,type,
aTP_Lamm_du: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__dv____,type,
aTP_Lamm_dv: fun_na7595719129879163854on_num ).
tff(sy_c_ATP_058Lamm__dw____,type,
aTP_Lamm_dw: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__dx____,type,
aTP_Lamm_dx: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__dy____,type,
aTP_Lamm_dy: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__dz____,type,
aTP_Lamm_dz: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__ea____,type,
aTP_Lamm_ea: code_integer > fun_Co3025358997185726883r_bool ).
tff(sy_c_ATP_058Lamm__eb____,type,
aTP_Lamm_eb: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__ec____,type,
aTP_Lamm_ec: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__ed____,type,
aTP_Lamm_ed: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__ee____,type,
aTP_Lamm_ee: nat > fun_na3408937105440547634t_bool ).
tff(sy_c_ATP_058Lamm__ef____,type,
aTP_Lamm_ef: fun_na4786964484885125043t_bool ).
tff(sy_c_ATP_058Lamm__eg____,type,
aTP_Lamm_eg: nat > fun_na3408937105440547634t_bool ).
tff(sy_c_ATP_058Lamm__eh____,type,
aTP_Lamm_eh: fun_na4786964484885125043t_bool ).
tff(sy_c_ATP_058Lamm__ei____,type,
aTP_Lamm_ei: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__ej____,type,
aTP_Lamm_ej: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__ek____,type,
aTP_Lamm_ek: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__el____,type,
aTP_Lamm_el: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__em____,type,
aTP_Lamm_em: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__en____,type,
aTP_Lamm_en: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__eo____,type,
aTP_Lamm_eo: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__ep____,type,
aTP_Lamm_ep: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__eq____,type,
aTP_Lamm_eq: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__er____,type,
aTP_Lamm_er: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__es____,type,
aTP_Lamm_es: int > fun_in6148020733644367646t_bool ).
tff(sy_c_ATP_058Lamm__et____,type,
aTP_Lamm_et: rat > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__eu____,type,
aTP_Lamm_eu: int > fun_in6148020733644367646t_bool ).
tff(sy_c_ATP_058Lamm__ev____,type,
aTP_Lamm_ev: rat > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__ew____,type,
aTP_Lamm_ew: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__ex____,type,
aTP_Lamm_ex: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__ey____,type,
aTP_Lamm_ey: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__ez____,type,
aTP_Lamm_ez: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__fa____,type,
aTP_Lamm_fa: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__fb____,type,
aTP_Lamm_fb: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__fc____,type,
aTP_Lamm_fc: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__fd____,type,
aTP_Lamm_fd: nat > fun_na5614638261385347865t_bool ).
tff(sy_c_ATP_058Lamm__fe____,type,
aTP_Lamm_fe: nat > fun_na5614638261385347865t_bool ).
tff(sy_c_ATP_058Lamm__ff____,type,
aTP_Lamm_ff: nat > fun_na5614638261385347865t_bool ).
tff(sy_c_ATP_058Lamm__fg____,type,
aTP_Lamm_fg: rat > fun_int_bool ).
tff(sy_c_ATP_058Lamm__fh____,type,
aTP_Lamm_fh: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__fi____,type,
aTP_Lamm_fi: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__fj____,type,
aTP_Lamm_fj: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__fk____,type,
aTP_Lamm_fk: fun_Pr803113274115125473t_bool ).
tff(sy_c_ATP_058Lamm__fl____,type,
aTP_Lamm_fl: fun_Pr1308109350235310159nt_int ).
tff(sy_c_ATP_058Lamm__fm____,type,
aTP_Lamm_fm: fun_Pr2478283820680522243nt_int ).
tff(sy_c_ATP_058Lamm__fn____,type,
aTP_Lamm_fn: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__fo____,type,
aTP_Lamm_fo: fun_Pr2478283820680522243nt_int ).
tff(sy_c_ATP_058Lamm__fp____,type,
aTP_Lamm_fp: fun_Pr1308109350235310159nt_int ).
tff(sy_c_ATP_058Lamm__fq____,type,
aTP_Lamm_fq: fun_int_int ).
tff(sy_c_ATP_058Lamm__fr____,type,
aTP_Lamm_fr: fun_num_fun_num_int ).
tff(sy_c_ATP_058Lamm__fs____,type,
aTP_Lamm_fs: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ft____,type,
aTP_Lamm_ft: fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__fu____,type,
aTP_Lamm_fu: nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__fv____,type,
aTP_Lamm_fv: nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__fw____,type,
aTP_Lamm_fw: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__fx____,type,
aTP_Lamm_fx: nat > fun_na5042103099923536755t_bool ).
tff(sy_c_ATP_058Lamm__fy____,type,
aTP_Lamm_fy: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__fz____,type,
aTP_Lamm_fz: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ga____,type,
aTP_Lamm_ga: rat > fun_Pr803113274115125473t_bool ).
tff(sy_c_ATP_058Lamm__gb____,type,
aTP_Lamm_gb: fun_na1476630412504255114at_nat ).
tff(sy_c_ATP_058Lamm__gc____,type,
aTP_Lamm_gc: nat > fun_na3408937105440547634t_bool ).
tff(sy_c_ATP_058Lamm__gd____,type,
aTP_Lamm_gd: fun_na4786964484885125043t_bool ).
tff(sy_c_ATP_058Lamm__ge____,type,
aTP_Lamm_ge: fun_nat_bool > fun_nat_bool ).
tff(sy_c_ATP_058Lamm__gf____,type,
aTP_Lamm_gf: set_int > fun_int_bool ).
tff(sy_c_ATP_058Lamm__gg____,type,
aTP_Lamm_gg: fun_nat_bool > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__gh____,type,
aTP_Lamm_gh: bool > fun_bo8333701921597992569l_char ).
tff(sy_c_ATP_058Lamm__gi____,type,
aTP_Lamm_gi: bool > fun_bo397793325678485071l_char ).
tff(sy_c_ATP_058Lamm__gj____,type,
aTP_Lamm_gj: bool > fun_bo8101500534245302755l_char ).
tff(sy_c_ATP_058Lamm__gk____,type,
aTP_Lamm_gk: bool > fun_bo252359628945711929l_char ).
tff(sy_c_ATP_058Lamm__gl____,type,
aTP_Lamm_gl: bool > fun_bo250947935745040205l_char ).
tff(sy_c_ATP_058Lamm__gm____,type,
aTP_Lamm_gm: bool > fun_bo8295548925090689571l_char ).
tff(sy_c_ATP_058Lamm__gn____,type,
aTP_Lamm_gn: bool > fun_Co4129272462331242167l_char ).
tff(sy_c_ATP_058Lamm__go____,type,
aTP_Lamm_go: fun_Co4129272462331242167l_char ).
tff(sy_c_ATP_058Lamm__gp____,type,
aTP_Lamm_gp: nat > fun_he5234484606919320760it_nat ).
tff(sy_c_ATP_058Lamm__gq____,type,
aTP_Lamm_gq: code_natural > fun_Co9168753396221475450atural ).
tff(sy_c_ATP_058Lamm__gr____,type,
aTP_Lamm_gr: produc7822875418678951345atural > fun_Co1890133979025130057atural ).
tff(sy_c_ATP_058Lamm__gs____,type,
aTP_Lamm_gs: code_natural > fun_Co1105491363547889651atural ).
tff(sy_c_ATP_058Lamm__gt____,type,
aTP_Lamm_gt: fun_Co1105491363547889651atural ).
tff(sy_c_ATP_058Lamm__gu____,type,
aTP_Lamm_gu: code_natural > fun_Co1105491363547889651atural ).
tff(sy_c_ATP_058Lamm__gv____,type,
aTP_Lamm_gv: fun_na819858167508000762st_nat ).
tff(sy_c_ATP_058Lamm__gw____,type,
aTP_Lamm_gw: fun_Pr2118491707810114563at_nat ).
tff(sy_c_ATP_058Lamm__gx____,type,
aTP_Lamm_gx: fun_Pr803113274115125473t_bool ).
tff(sy_c_ATP_058Lamm__gy____,type,
aTP_Lamm_gy: fun_se3032438776331272065t_bool ).
tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
archim2889992004027027881ng_rat: fun_rat_int ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
archim3151403230148437115or_rat: fun_rat_int ).
tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
archimedean_frac_rat: fun_rat_rat ).
tff(sy_c_Array__Time_Omake_001t__HOL__Obool,type,
array_make_bool: ( nat * fun_nat_bool ) > heap_T6374562886178639292y_bool ).
tff(sy_c_Array__Time_Omake_001t__Int__Oint,type,
array_make_int: ( nat * fun_nat_int ) > heap_T1346037964561226099ay_int ).
tff(sy_c_Array__Time_Omake_001t__Nat__Onat,type,
array_make_nat: ( nat * fun_nat_nat ) > heap_T3836121109492952855ay_nat ).
tff(sy_c_Array__Time_Onth_001t__Product____Type__Ounit,type,
array_7872002506669749220t_unit: ( array_Product_unit * nat ) > heap_T5738788834812785303t_unit ).
tff(sy_c_Array__Time_Oof__list_001t__HOL__Obool,type,
array_of_list_bool: list_bool > heap_T6374562886178639292y_bool ).
tff(sy_c_Array__Time_Oof__list_001t__Int__Oint,type,
array_of_list_int: list_int > heap_T1346037964561226099ay_int ).
tff(sy_c_Array__Time_Oof__list_001t__Nat__Onat,type,
array_of_list_nat: list_nat > heap_T3836121109492952855ay_nat ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__HOL__Obool_001t__HOL__Obool,type,
bNF_re3324582151780938536l_bool: ( fun_in5519481650651148416r_bool * fun_bo1841576699660343469l_bool ) > fun_fu2005298298577998622l_bool ).
tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re3379532845092657523nteger: ( fun_in5519481650651148416r_bool * fun_in5519481650651148416r_bool ) > fun_fu6750301411456490045r_bool ).
tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint,type,
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member_list_nat: ( list_nat * set_list_nat ) > $o ).
tff(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > fun_set_nat_bool ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).
tff(sy_c_member_001t__Rat__Orat,type,
member_rat: ( rat * set_rat ) > $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__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member2340774599025711042nt_int: ( set_Pr958786334691620121nt_int * set_se6260736226359567993nt_int ) > $o ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_P,type,
p: bool ).
tff(sy_v_h,type,
h: heap_e7401611519738050253t_unit ).
tff(sy_v_m,type,
m: heap_Time_Heap_a ).
tff(sy_v_t,type,
t: nat ).
% Relevant facts (6253)
tff(fact_0__092_060open_062_092_060And_062h_O_AP_A_092_060Longrightarrow_062_Atime_Am_Ah_A_092_060le_062_At_092_060close_062,axiom,
! [H: heap_e7401611519738050253t_unit] :
( pp(p)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_a(m,H)),t)) ) ).
% \<open>\<And>h. P \<Longrightarrow> time m h \<le> t\<close>
tff(fact_1_assms,axiom,
( pp(p)
=> pp(aa_nat_bool(time_TBOUND_a(m),t)) ) ).
% assms
tff(fact_2_Heap__Time__Monad_Obind__bind,axiom,
! [F: heap_T5738788834812785303t_unit,G: fun_Pr5215307506330550056Heap_a,K: fun_a_2990225006182978344t_unit] : heap_T8685611227969916822t_unit(heap_T757603679106148408unit_a(F,G),K) = heap_T2633723481742716231t_unit(F,aa_fun558819986861788875t_unit(aTP_Lamm_aa(G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_3_Heap__Time__Monad_Obind__bind,axiom,
! [F: heap_Time_Heap_a,G: fun_a_2990225006182978344t_unit,K: fun_Pr5215307506330550056Heap_a] : heap_T757603679106148408unit_a(heap_T8685611227969916822t_unit(F,G),K) = heap_Time_bind_a_a(F,aa_fun2786189307152378797Heap_a(aTP_Lamm_ab(G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_4_Heap__Time__Monad_Obind__bind,axiom,
! [F: heap_T5738788834812785303t_unit,G: fun_Pr292408699144966967t_unit,K: fun_Pr292408699144966967t_unit] : heap_T2633723481742716231t_unit(heap_T2633723481742716231t_unit(F,G),K) = heap_T2633723481742716231t_unit(F,aa_fun7583176353580103804t_unit(aTP_Lamm_ac(G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_5_Heap__Time__Monad_Obind__bind,axiom,
! [F: heap_Time_Heap_a,G: fun_a_6093291656547882009Heap_a,K: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(heap_Time_bind_a_a(F,G),K) = heap_Time_bind_a_a(F,aa_fun5994237636032447228Heap_a(aTP_Lamm_ad(G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_6_Heap__Time__Monad_Obind__bind,axiom,
! [F: heap_T5738788834812785303t_unit,G: fun_Pr5215307506330550056Heap_a,K: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(heap_T757603679106148408unit_a(F,G),K) = heap_T757603679106148408unit_a(F,aa_fun7182454221404266251Heap_a(aTP_Lamm_ae(G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_7_Heap__Time__Monad_Obind__bind,axiom,
! [F: heap_T5738788834812785303t_unit,G: fun_Pr292408699144966967t_unit,K: fun_Pr5215307506330550056Heap_a] : heap_T757603679106148408unit_a(heap_T2633723481742716231t_unit(F,G),K) = heap_T757603679106148408unit_a(F,aa_fun8407955803497444028Heap_a(aTP_Lamm_af(G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_8_order__refl,axiom,
! [X: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X),X)) ).
% order_refl
tff(fact_9_order__refl,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).
% order_refl
tff(fact_10_order__refl,axiom,
! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).
% order_refl
tff(fact_11_order__refl,axiom,
! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).
% order_refl
tff(fact_12_order__refl,axiom,
! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).
% order_refl
tff(fact_13_dual__order_Orefl,axiom,
! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).
% dual_order.refl
tff(fact_14_dual__order_Orefl,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).
% dual_order.refl
tff(fact_15_dual__order_Orefl,axiom,
! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).
% dual_order.refl
tff(fact_16_dual__order_Orefl,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).
% dual_order.refl
tff(fact_17_dual__order_Orefl,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).
% dual_order.refl
tff(fact_18_distrib__if__bind,axiom,
! [B: bool,C: heap_T5738788834812785303t_unit,D: heap_T5738788834812785303t_unit,F: fun_Pr292408699144966967t_unit] :
( ( pp(B)
=> ( heap_T2633723481742716231t_unit(if_Hea8138950348631371857t_unit(B,C,D),F) = heap_T2633723481742716231t_unit(C,F) ) )
& ( ~ pp(B)
=> ( heap_T2633723481742716231t_unit(if_Hea8138950348631371857t_unit(B,C,D),F) = heap_T2633723481742716231t_unit(D,F) ) ) ) ).
% distrib_if_bind
tff(fact_19_distrib__if__bind,axiom,
! [B: bool,C: heap_Time_Heap_a,D: heap_Time_Heap_a,F: fun_a_6093291656547882009Heap_a] :
( ( pp(B)
=> ( heap_Time_bind_a_a(if_Heap_Time_Heap_a(B,C,D),F) = heap_Time_bind_a_a(C,F) ) )
& ( ~ pp(B)
=> ( heap_Time_bind_a_a(if_Heap_Time_Heap_a(B,C,D),F) = heap_Time_bind_a_a(D,F) ) ) ) ).
% distrib_if_bind
tff(fact_20_distrib__if__bind,axiom,
! [B: bool,C: heap_T5738788834812785303t_unit,D: heap_T5738788834812785303t_unit,F: fun_Pr5215307506330550056Heap_a] :
( ( pp(B)
=> ( heap_T757603679106148408unit_a(if_Hea8138950348631371857t_unit(B,C,D),F) = heap_T757603679106148408unit_a(C,F) ) )
& ( ~ pp(B)
=> ( heap_T757603679106148408unit_a(if_Hea8138950348631371857t_unit(B,C,D),F) = heap_T757603679106148408unit_a(D,F) ) ) ) ).
% distrib_if_bind
tff(fact_21_TBOUNDD,axiom,
! [M: heap_T5738788834812785303t_unit,T: nat,H: heap_e7401611519738050253t_unit] :
( pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H)),T)) ) ).
% TBOUNDD
tff(fact_22_TBOUNDD,axiom,
! [M: heap_Time_Heap_a,T: nat,H: heap_e7401611519738050253t_unit] :
( pp(aa_nat_bool(time_TBOUND_a(M),T))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_a(M,H)),T)) ) ).
% TBOUNDD
tff(fact_23_TBOUNDI,axiom,
! [M: heap_T5738788834812785303t_unit,T: nat] :
( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H2)),T))
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T)) ) ).
% TBOUNDI
tff(fact_24_TBOUNDI,axiom,
! [M: heap_Time_Heap_a,T: nat] :
( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_a(M,H2)),T))
=> pp(aa_nat_bool(time_TBOUND_a(M),T)) ) ).
% TBOUNDI
tff(fact_25_TBOUND__def,axiom,
! [M: heap_T5738788834812785303t_unit,T: nat] :
( pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T))
<=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H3)),T)) ) ).
% TBOUND_def
tff(fact_26_TBOUND__def,axiom,
! [M: heap_Time_Heap_a,T: nat] :
( pp(aa_nat_bool(time_TBOUND_a(M),T))
<=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_a(M,H3)),T)) ) ).
% TBOUND_def
tff(fact_27_le__refl,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).
% le_refl
tff(fact_28_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).
% le_trans
tff(fact_29_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% eq_imp_le
tff(fact_30_le__antisym,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( M = N ) ) ) ).
% le_antisym
tff(fact_31_nat__le__linear,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).
% nat_le_linear
tff(fact_32_TBOUND__mono,axiom,
! [C: heap_Time_Heap_a,T: nat,T2: nat] :
( pp(aa_nat_bool(time_TBOUND_a(C),T))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T2))
=> pp(aa_nat_bool(time_TBOUND_a(C),T2)) ) ) ).
% TBOUND_mono
tff(fact_33_TBOUND__mono,axiom,
! [C: heap_T5738788834812785303t_unit,T: nat,T2: nat] :
( pp(aa_nat_bool(time_T7469515765551943773t_unit(C),T))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T2))
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(C),T2)) ) ) ).
% TBOUND_mono
tff(fact_34_TBOUND__eqI,axiom,
! [M: heap_Time_Heap_a,T: nat] :
( ! [H2: heap_e7401611519738050253t_unit] : time_time_a(M,H2) = T
=> pp(aa_nat_bool(time_TBOUND_a(M),T)) ) ).
% TBOUND_eqI
tff(fact_35_TBOUND__eqI,axiom,
! [M: heap_T5738788834812785303t_unit,T: nat] :
( ! [H2: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(M,H2) = T
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T)) ) ).
% TBOUND_eqI
tff(fact_36_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_37_order__antisym__conv,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_38_order__antisym__conv,axiom,
! [Y: num,X: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_39_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_40_order__antisym__conv,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_41_linorder__le__cases,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).
% linorder_le_cases
tff(fact_42_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).
% linorder_le_cases
tff(fact_43_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).
% linorder_le_cases
tff(fact_44_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).
% linorder_le_cases
tff(fact_45_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ( aa_rat_rat(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_46_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ( aa_rat_num(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_47_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ( aa_rat_nat(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_48_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ( aa_rat_int(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_49_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( ( aa_num_rat(F,B) = C )
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_50_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( ( aa_num_num(F,B) = C )
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_51_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_nat,C: nat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( ( aa_num_nat(F,B) = C )
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_52_ord__le__eq__subst,axiom,
! [A: num,B: num,F: fun_num_int,C: int] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( ( aa_num_int(F,B) = C )
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_53_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( ( aa_nat_rat(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_54_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_num,C: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( ( aa_nat_num(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).
% ord_le_eq_subst
tff(fact_55_ord__eq__le__subst,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( ( A = aa_rat_rat(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_56_ord__eq__le__subst,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( ( A = aa_rat_num(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_57_ord__eq__le__subst,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( ( A = aa_rat_nat(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_58_ord__eq__le__subst,axiom,
! [A: int,F: fun_rat_int,B: rat,C: rat] :
( ( A = aa_rat_int(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_59_ord__eq__le__subst,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( ( A = aa_num_rat(F,B) )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_60_ord__eq__le__subst,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( ( A = aa_num_num(F,B) )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_61_ord__eq__le__subst,axiom,
! [A: nat,F: fun_num_nat,B: num,C: num] :
( ( A = aa_num_nat(F,B) )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_62_ord__eq__le__subst,axiom,
! [A: int,F: fun_num_int,B: num,C: num] :
( ( A = aa_num_int(F,B) )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_63_ord__eq__le__subst,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( ( A = aa_nat_rat(F,B) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_64_ord__eq__le__subst,axiom,
! [A: num,F: fun_nat_num,B: nat,C: nat] :
( ( A = aa_nat_num(F,B) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).
% ord_eq_le_subst
tff(fact_65_linorder__linear,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).
% linorder_linear
tff(fact_66_linorder__linear,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).
% linorder_linear
tff(fact_67_linorder__linear,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).
% linorder_linear
tff(fact_68_linorder__linear,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).
% linorder_linear
tff(fact_69_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).
% order_eq_refl
tff(fact_70_order__eq__refl,axiom,
! [X: rat,Y: rat] :
( ( X = Y )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% order_eq_refl
tff(fact_71_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).
% order_eq_refl
tff(fact_72_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).
% order_eq_refl
tff(fact_73_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).
% order_eq_refl
tff(fact_74_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_75_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_76_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_77_order__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_78_order__subst2,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_79_order__subst2,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_80_order__subst2,axiom,
! [A: num,B: num,F: fun_num_nat,C: nat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_81_order__subst2,axiom,
! [A: num,B: num,F: fun_num_int,C: int] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_82_order__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_83_order__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_num,C: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B)),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).
% order_subst2
tff(fact_84_order__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).
% order_subst1
tff(fact_85_order__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).
% order_subst1
tff(fact_86_order__subst1,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).
% order_subst1
tff(fact_87_order__subst1,axiom,
! [A: rat,F: fun_int_rat,B: int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C))) ) ) ) ).
% order_subst1
tff(fact_88_order__subst1,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).
% order_subst1
tff(fact_89_order__subst1,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).
% order_subst1
tff(fact_90_order__subst1,axiom,
! [A: num,F: fun_nat_num,B: nat,C: nat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).
% order_subst1
tff(fact_91_order__subst1,axiom,
! [A: num,F: fun_int_num,B: int,C: int] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X2)),aa_int_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C))) ) ) ) ).
% order_subst1
tff(fact_92_order__subst1,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).
% order_subst1
tff(fact_93_order__subst1,axiom,
! [A: nat,F: fun_num_nat,B: num,C: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).
% order_subst1
tff(fact_94_Orderings_Oorder__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).
% Orderings.order_eq_iff
tff(fact_95_Orderings_Oorder__eq__iff,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% Orderings.order_eq_iff
tff(fact_96_Orderings_Oorder__eq__iff,axiom,
! [A: num,B: num] :
( ( A = B )
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).
% Orderings.order_eq_iff
tff(fact_97_Orderings_Oorder__eq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).
% Orderings.order_eq_iff
tff(fact_98_Orderings_Oorder__eq__iff,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).
% Orderings.order_eq_iff
tff(fact_99_antisym,axiom,
! [A: set_nat,B: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
=> ( A = B ) ) ) ).
% antisym
tff(fact_100_antisym,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( A = B ) ) ) ).
% antisym
tff(fact_101_antisym,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( A = B ) ) ) ).
% antisym
tff(fact_102_antisym,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( A = B ) ) ) ).
% antisym
tff(fact_103_antisym,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( A = B ) ) ) ).
% antisym
tff(fact_104_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A)) ) ) ).
% dual_order.trans
tff(fact_105_dual__order_Otrans,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).
% dual_order.trans
tff(fact_106_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).
% dual_order.trans
tff(fact_107_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).
% dual_order.trans
tff(fact_108_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).
% dual_order.trans
tff(fact_109_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_110_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_111_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_112_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_113_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_114_dual__order_Oeq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).
% dual_order.eq_iff
tff(fact_115_dual__order_Oeq__iff,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% dual_order.eq_iff
tff(fact_116_dual__order_Oeq__iff,axiom,
! [A: num,B: num] :
( ( A = B )
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).
% dual_order.eq_iff
tff(fact_117_dual__order_Oeq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% dual_order.eq_iff
tff(fact_118_dual__order_Oeq__iff,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% dual_order.eq_iff
tff(fact_119_linorder__wlog,axiom,
! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
( ! [A2: rat,B2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A2),B2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B2)) )
=> ( ! [A2: rat,B2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B2),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B2)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ).
% linorder_wlog
tff(fact_120_linorder__wlog,axiom,
! [P: fun_num_fun_num_bool,A: num,B: num] :
( ! [A2: num,B2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
=> pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) )
=> ( ! [A2: num,B2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(P,B2),A2))
=> pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) )
=> pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ).
% linorder_wlog
tff(fact_121_linorder__wlog,axiom,
! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) )
=> ( ! [A2: nat,B2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B2),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ).
% linorder_wlog
tff(fact_122_linorder__wlog,axiom,
! [P: fun_int_fun_int_bool,A: int,B: int] :
( ! [A2: int,B2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) )
=> ( ! [A2: int,B2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(P,B2),A2))
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ).
% linorder_wlog
tff(fact_123_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Z)) ) ) ).
% order_trans
tff(fact_124_order__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) ) ) ).
% order_trans
tff(fact_125_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) ) ) ).
% order_trans
tff(fact_126_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).
% order_trans
tff(fact_127_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).
% order_trans
tff(fact_128_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).
% order.trans
tff(fact_129_order_Otrans,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).
% order.trans
tff(fact_130_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).
% order.trans
tff(fact_131_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).
% order.trans
tff(fact_132_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).
% order.trans
tff(fact_133_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_134_order__antisym,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_135_order__antisym,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_136_order__antisym,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_137_order__antisym,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_138_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> ( ( B = C )
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).
% ord_le_eq_trans
tff(fact_139_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ( B = C )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).
% ord_le_eq_trans
tff(fact_140_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( ( B = C )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).
% ord_le_eq_trans
tff(fact_141_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( ( B = C )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).
% ord_le_eq_trans
tff(fact_142_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( ( B = C )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).
% ord_le_eq_trans
tff(fact_143_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).
% ord_eq_le_trans
tff(fact_144_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).
% ord_eq_le_trans
tff(fact_145_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).
% ord_eq_le_trans
tff(fact_146_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).
% ord_eq_le_trans
tff(fact_147_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).
% ord_eq_le_trans
tff(fact_148_order__class_Oorder__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).
% order_class.order_eq_iff
tff(fact_149_order__class_Oorder__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( X = Y )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).
% order_class.order_eq_iff
tff(fact_150_order__class_Oorder__eq__iff,axiom,
! [X: num,Y: num] :
( ( X = Y )
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).
% order_class.order_eq_iff
tff(fact_151_order__class_Oorder__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).
% order_class.order_eq_iff
tff(fact_152_order__class_Oorder__eq__iff,axiom,
! [X: int,Y: int] :
( ( X = Y )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).
% order_class.order_eq_iff
tff(fact_153_le__cases3,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) )
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) )
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) )
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X)) )
=> ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ) ) ) ) ).
% le_cases3
tff(fact_154_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z)) )
=> ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) )
=> ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) )
=> ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) )
=> ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X)) )
=> ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ) ) ) ) ) ).
% le_cases3
tff(fact_155_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X)) )
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).
% le_cases3
tff(fact_156_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X)) )
=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).
% le_cases3
tff(fact_157_nle__le,axiom,
! [A: rat,B: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
& ( B != A ) ) ) ).
% nle_le
tff(fact_158_nle__le,axiom,
! [A: num,B: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
& ( B != A ) ) ) ).
% nle_le
tff(fact_159_nle__le,axiom,
! [A: nat,B: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
& ( B != A ) ) ) ).
% nle_le
tff(fact_160_nle__le,axiom,
! [A: int,B: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& ( B != A ) ) ) ).
% nle_le
tff(fact_161_Nat_Oex__has__greatest__nat,axiom,
! [P: fun_nat_bool,K: nat,B: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> ? [X2: nat] :
( pp(aa_nat_bool(P,X2))
& ! [Y3: nat] :
( pp(aa_nat_bool(P,Y3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) ) ) ) ) ).
% Nat.ex_has_greatest_nat
tff(fact_162_raise__bind,axiom,
! [E: list_char,F: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(heap_Time_raise_a(E),F) = heap_Time_raise_a(E) ).
% raise_bind
tff(fact_163_raise__bind,axiom,
! [E: list_char,F: fun_Pr5215307506330550056Heap_a] : heap_T757603679106148408unit_a(heap_T2927564422264180874t_unit(E),F) = heap_Time_raise_a(E) ).
% raise_bind
tff(fact_164_raise__bind,axiom,
! [E: list_char,F: fun_Pr292408699144966967t_unit] : heap_T2633723481742716231t_unit(heap_T2927564422264180874t_unit(E),F) = heap_T2927564422264180874t_unit(E) ).
% raise_bind
tff(fact_165_Greatest__equality,axiom,
! [P: fun_set_nat_bool,X: set_nat] :
( pp(aa_set_nat_bool(P,X))
=> ( ! [Y2: set_nat] :
( pp(aa_set_nat_bool(P,Y2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y2),X)) )
=> ( order_5724808138429204845et_nat(P) = X ) ) ) ).
% Greatest_equality
tff(fact_166_Greatest__equality,axiom,
! [P: fun_rat_bool,X: rat] :
( pp(aa_rat_bool(P,X))
=> ( ! [Y2: rat] :
( pp(aa_rat_bool(P,Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y2),X)) )
=> ( order_Greatest_rat(P) = X ) ) ) ).
% Greatest_equality
tff(fact_167_Greatest__equality,axiom,
! [P: fun_num_bool,X: num] :
( pp(aa_num_bool(P,X))
=> ( ! [Y2: num] :
( pp(aa_num_bool(P,Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y2),X)) )
=> ( order_Greatest_num(P) = X ) ) ) ).
% Greatest_equality
tff(fact_168_Greatest__equality,axiom,
! [P: fun_int_bool,X: int] :
( pp(aa_int_bool(P,X))
=> ( ! [Y2: int] :
( pp(aa_int_bool(P,Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y2),X)) )
=> ( order_Greatest_int(P) = X ) ) ) ).
% Greatest_equality
tff(fact_169_Greatest__equality,axiom,
! [P: fun_nat_bool,X: nat] :
( pp(aa_nat_bool(P,X))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),X)) )
=> ( order_Greatest_nat(P) = X ) ) ) ).
% Greatest_equality
tff(fact_170_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
( member8440522571783428010at_nat(A,collec3392354462482085612at_nat(P))
<=> pp(aa_Pro2232559814647311250t_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_171_mem__Collect__eq,axiom,
! [A: list_nat,P: fun_list_nat_bool] :
( member_list_nat(A,collect_list_nat(P))
<=> pp(aa_list_nat_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_172_mem__Collect__eq,axiom,
! [A: set_Pr958786334691620121nt_int,P: fun_se3032438776331272065t_bool] :
( member2340774599025711042nt_int(A,collec5210948495886036740nt_int(P))
<=> pp(aa_set7317184552576006522t_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_173_mem__Collect__eq,axiom,
! [A: set_nat,P: fun_set_nat_bool] :
( member_set_nat(A,collect_set_nat(P))
<=> pp(aa_set_nat_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_174_mem__Collect__eq,axiom,
! [A: nat,P: fun_nat_bool] :
( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
<=> pp(aa_nat_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_175_mem__Collect__eq,axiom,
! [A: int,P: fun_int_bool] :
( member_int(A,collect_int(P))
<=> pp(aa_int_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_176_Collect__mem__eq,axiom,
! [A3: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_ag(A3)) = A3 ).
% Collect_mem_eq
tff(fact_177_Collect__mem__eq,axiom,
! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ah(A3)) = A3 ).
% Collect_mem_eq
tff(fact_178_Collect__mem__eq,axiom,
! [A3: set_se6260736226359567993nt_int] : collec5210948495886036740nt_int(aTP_Lamm_ai(A3)) = A3 ).
% Collect_mem_eq
tff(fact_179_Collect__mem__eq,axiom,
! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_aj(A3)) = A3 ).
% Collect_mem_eq
tff(fact_180_Collect__mem__eq,axiom,
! [A3: set_nat] : collect_nat(aTP_Lamm_ak(A3)) = A3 ).
% Collect_mem_eq
tff(fact_181_Collect__mem__eq,axiom,
! [A3: set_int] : collect_int(aTP_Lamm_al(A3)) = A3 ).
% Collect_mem_eq
tff(fact_182_Collect__cong,axiom,
! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
( ! [X2: list_nat] :
( pp(aa_list_nat_bool(P,X2))
<=> pp(aa_list_nat_bool(Q,X2)) )
=> ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).
% Collect_cong
tff(fact_183_Collect__cong,axiom,
! [P: fun_se3032438776331272065t_bool,Q: fun_se3032438776331272065t_bool] :
( ! [X2: set_Pr958786334691620121nt_int] :
( pp(aa_set7317184552576006522t_bool(P,X2))
<=> pp(aa_set7317184552576006522t_bool(Q,X2)) )
=> ( collec5210948495886036740nt_int(P) = collec5210948495886036740nt_int(Q) ) ) ).
% Collect_cong
tff(fact_184_Collect__cong,axiom,
! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
( ! [X2: set_nat] :
( pp(aa_set_nat_bool(P,X2))
<=> pp(aa_set_nat_bool(Q,X2)) )
=> ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).
% Collect_cong
tff(fact_185_Collect__cong,axiom,
! [P: fun_nat_bool,Q: fun_nat_bool] :
( ! [X2: nat] :
( pp(aa_nat_bool(P,X2))
<=> pp(aa_nat_bool(Q,X2)) )
=> ( collect_nat(P) = collect_nat(Q) ) ) ).
% Collect_cong
tff(fact_186_Collect__cong,axiom,
! [P: fun_int_bool,Q: fun_int_bool] :
( ! [X2: int] :
( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(Q,X2)) )
=> ( collect_int(P) = collect_int(Q) ) ) ).
% Collect_cong
tff(fact_187_GreatestI2__order,axiom,
! [P: fun_set_nat_bool,X: set_nat,Q: fun_set_nat_bool] :
( pp(aa_set_nat_bool(P,X))
=> ( ! [Y2: set_nat] :
( pp(aa_set_nat_bool(P,Y2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y2),X)) )
=> ( ! [X2: set_nat] :
( pp(aa_set_nat_bool(P,X2))
=> ( ! [Y3: set_nat] :
( pp(aa_set_nat_bool(P,Y3))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y3),X2)) )
=> pp(aa_set_nat_bool(Q,X2)) ) )
=> pp(aa_set_nat_bool(Q,order_5724808138429204845et_nat(P))) ) ) ) ).
% GreatestI2_order
tff(fact_188_GreatestI2__order,axiom,
! [P: fun_rat_bool,X: rat,Q: fun_rat_bool] :
( pp(aa_rat_bool(P,X))
=> ( ! [Y2: rat] :
( pp(aa_rat_bool(P,Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y2),X)) )
=> ( ! [X2: rat] :
( pp(aa_rat_bool(P,X2))
=> ( ! [Y3: rat] :
( pp(aa_rat_bool(P,Y3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X2)) )
=> pp(aa_rat_bool(Q,X2)) ) )
=> pp(aa_rat_bool(Q,order_Greatest_rat(P))) ) ) ) ).
% GreatestI2_order
tff(fact_189_GreatestI2__order,axiom,
! [P: fun_num_bool,X: num,Q: fun_num_bool] :
( pp(aa_num_bool(P,X))
=> ( ! [Y2: num] :
( pp(aa_num_bool(P,Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y2),X)) )
=> ( ! [X2: num] :
( pp(aa_num_bool(P,X2))
=> ( ! [Y3: num] :
( pp(aa_num_bool(P,Y3))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X2)) )
=> pp(aa_num_bool(Q,X2)) ) )
=> pp(aa_num_bool(Q,order_Greatest_num(P))) ) ) ) ).
% GreatestI2_order
tff(fact_190_GreatestI2__order,axiom,
! [P: fun_int_bool,X: int,Q: fun_int_bool] :
( pp(aa_int_bool(P,X))
=> ( ! [Y2: int] :
( pp(aa_int_bool(P,Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y2),X)) )
=> ( ! [X2: int] :
( pp(aa_int_bool(P,X2))
=> ( ! [Y3: int] :
( pp(aa_int_bool(P,Y3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X2)) )
=> pp(aa_int_bool(Q,X2)) ) )
=> pp(aa_int_bool(Q,order_Greatest_int(P))) ) ) ) ).
% GreatestI2_order
tff(fact_191_GreatestI2__order,axiom,
! [P: fun_nat_bool,X: nat,Q: fun_nat_bool] :
( pp(aa_nat_bool(P,X))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),X)) )
=> ( ! [X2: nat] :
( pp(aa_nat_bool(P,X2))
=> ( ! [Y3: nat] :
( pp(aa_nat_bool(P,Y3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X2)) )
=> pp(aa_nat_bool(Q,X2)) ) )
=> pp(aa_nat_bool(Q,order_Greatest_nat(P))) ) ) ) ).
% GreatestI2_order
tff(fact_192_TBOUND__assert_H__bind,axiom,
! [P: bool,P2: bool,M: heap_Time_Heap_a,T: nat] :
( time_EQ_bool(P,P2)
=> ( ( pp(P)
=> pp(aa_nat_bool(time_TBOUND_a(M),T)) )
=> pp(aa_nat_bool(time_TBOUND_a(heap_T757603679106148408unit_a(refine_Imp_assert(P),aTP_Lamm_am(M))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P2),T),zero_zero_nat))) ) ) ).
% TBOUND_assert'_bind
tff(fact_193_TBOUND__assert_H__bind,axiom,
! [P: bool,P2: bool,M: heap_T5738788834812785303t_unit,T: nat] :
( time_EQ_bool(P,P2)
=> ( ( pp(P)
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T)) )
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(heap_T2633723481742716231t_unit(refine_Imp_assert(P),aTP_Lamm_an(M))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P2),T),zero_zero_nat))) ) ) ).
% TBOUND_assert'_bind
tff(fact_194_bounded__Max__nat,axiom,
! [P: fun_nat_bool,X: nat,M2: nat] :
( pp(aa_nat_bool(P,X))
=> ( ! [X2: nat] :
( pp(aa_nat_bool(P,X2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M2)) )
=> ~ ! [M3: nat] :
( pp(aa_nat_bool(P,M3))
=> ~ ! [X3: nat] :
( pp(aa_nat_bool(P,X3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M3)) ) ) ) ) ).
% bounded_Max_nat
tff(fact_195_le__rel__bool__arg__iff,axiom,
! [X4: fun_bool_set_nat,Y4: fun_bool_set_nat] :
( ord_le6215889286817668944et_nat(X4,Y4)
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_bool_set_nat(X4,fFalse)),aa_bool_set_nat(Y4,fFalse)))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_bool_set_nat(X4,fTrue)),aa_bool_set_nat(Y4,fTrue))) ) ) ).
% le_rel_bool_arg_iff
tff(fact_196_le__rel__bool__arg__iff,axiom,
! [X4: fun_bool_rat,Y4: fun_bool_rat] :
( ord_le7121926485642632354ol_rat(X4,Y4)
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(X4,fFalse)),aa_bool_rat(Y4,fFalse)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(X4,fTrue)),aa_bool_rat(Y4,fTrue))) ) ) ).
% le_rel_bool_arg_iff
tff(fact_197_le__rel__bool__arg__iff,axiom,
! [X4: fun_bool_num,Y4: fun_bool_num] :
( ord_le2115026944130090980ol_num(X4,Y4)
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_bool_num(X4,fFalse)),aa_bool_num(Y4,fFalse)))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_bool_num(X4,fTrue)),aa_bool_num(Y4,fTrue))) ) ) ).
% le_rel_bool_arg_iff
tff(fact_198_le__rel__bool__arg__iff,axiom,
! [X4: fun_bool_nat,Y4: fun_bool_nat] :
( ord_le1406783007181253274ol_nat(X4,Y4)
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(X4,fFalse)),aa_bool_nat(Y4,fFalse)))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(X4,fTrue)),aa_bool_nat(Y4,fTrue))) ) ) ).
% le_rel_bool_arg_iff
tff(fact_199_le__rel__bool__arg__iff,axiom,
! [X4: fun_bool_int,Y4: fun_bool_int] :
( ord_le6452304024526832374ol_int(X4,Y4)
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(X4,fFalse)),aa_bool_int(Y4,fFalse)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(X4,fTrue)),aa_bool_int(Y4,fTrue))) ) ) ).
% le_rel_bool_arg_iff
tff(fact_200_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
| ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% verit_la_disequality
tff(fact_201_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
| ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).
% verit_la_disequality
tff(fact_202_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).
% verit_la_disequality
tff(fact_203_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
| ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% verit_la_disequality
tff(fact_204_ord__eq__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( A = B )
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
=> ( ( C = D )
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),D)) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_205_ord__eq__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( A = B )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ( C = D )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),D)) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_206_ord__eq__le__eq__trans,axiom,
! [A: num,B: num,C: num,D: num] :
( ( A = B )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ( C = D )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),D)) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_207_ord__eq__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A = B )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( ( C = D )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),D)) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_208_ord__eq__le__eq__trans,axiom,
! [A: int,B: int,C: int,D: int] :
( ( A = B )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> ( ( C = D )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),D)) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_209_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).
% verit_comp_simplify1(2)
tff(fact_210_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_211_verit__comp__simplify1_I2_J,axiom,
! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_212_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_213_verit__comp__simplify1_I2_J,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_214_refines__assert_H__bind,axiom,
! [P3: heap_Time_Heap_a,Q2: heap_Time_Heap_a,Phi: bool] :
( refine_Imp_refines_a(P3,Q2)
=> refine_Imp_refines_a(P3,heap_T757603679106148408unit_a(refine_Imp_assert(Phi),aTP_Lamm_am(Q2))) ) ).
% refines_assert'_bind
tff(fact_215_refines__assert_H__bind,axiom,
! [P3: heap_T5738788834812785303t_unit,Q2: heap_T5738788834812785303t_unit,Phi: bool] :
( refine451278484176762712t_unit(P3,Q2)
=> refine451278484176762712t_unit(P3,heap_T2633723481742716231t_unit(refine_Imp_assert(Phi),aTP_Lamm_an(Q2))) ) ).
% refines_assert'_bind
tff(fact_216_ureturn__bind,axiom,
! [X: a,F: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,X),F) = aa_a_H8829438773920479562Heap_a(F,X) ).
% ureturn_bind
tff(fact_217_ureturn__bind,axiom,
! [X: product_unit,F: fun_Pr5215307506330550056Heap_a] : heap_T757603679106148408unit_a(aa_Pro1108561329362217960t_unit(heap_T4284346855313245393t_unit,X),F) = aa_Pro7886978635576167513Heap_a(F,X) ).
% ureturn_bind
tff(fact_218_ureturn__bind,axiom,
! [X: product_unit,F: fun_Pr292408699144966967t_unit] : heap_T2633723481742716231t_unit(aa_Pro1108561329362217960t_unit(heap_T4284346855313245393t_unit,X),F) = aa_Pro1108561329362217960t_unit(F,X) ).
% ureturn_bind
tff(fact_219_bot__nat__0_Oextremum,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).
% bot_nat_0.extremum
tff(fact_220_le0,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
% le0
tff(fact_221_refines__refl,axiom,
! [P3: heap_T5738788834812785303t_unit] : refine451278484176762712t_unit(P3,P3) ).
% refines_refl
tff(fact_222_refines__If,axiom,
! [B: bool,T: heap_T5738788834812785303t_unit,T2: heap_T5738788834812785303t_unit,E: heap_T5738788834812785303t_unit,E2: heap_T5738788834812785303t_unit] :
( ( pp(B)
=> refine451278484176762712t_unit(T,T2) )
=> ( ( ~ pp(B)
=> refine451278484176762712t_unit(E,E2) )
=> refine451278484176762712t_unit(if_Hea8138950348631371857t_unit(B,T,E),if_Hea8138950348631371857t_unit(B,T2,E2)) ) ) ).
% refines_If
tff(fact_223_time__raise,axiom,
! [Msg: list_char,H: heap_e7401611519738050253t_unit] : time_time_a(heap_Time_raise_a(Msg),H) = zero_zero_nat ).
% time_raise
tff(fact_224_time__raise,axiom,
! [Msg: list_char,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(heap_T2927564422264180874t_unit(Msg),H) = zero_zero_nat ).
% time_raise
tff(fact_225_refines__bind,axiom,
! [M: heap_Time_Heap_a,M4: heap_Time_Heap_a,F: fun_a_6093291656547882009Heap_a,F2: fun_a_6093291656547882009Heap_a] :
( refine_Imp_refines_a(M,M4)
=> ( ! [X2: a] : refine_Imp_refines_a(aa_a_H8829438773920479562Heap_a(F,X2),aa_a_H8829438773920479562Heap_a(F2,X2))
=> refine_Imp_refines_a(heap_Time_bind_a_a(M,F),heap_Time_bind_a_a(M4,F2)) ) ) ).
% refines_bind
tff(fact_226_refines__bind,axiom,
! [M: heap_T5738788834812785303t_unit,M4: heap_T5738788834812785303t_unit,F: fun_Pr5215307506330550056Heap_a,F2: fun_Pr5215307506330550056Heap_a] :
( refine451278484176762712t_unit(M,M4)
=> ( ! [X2: product_unit] : refine_Imp_refines_a(aa_Pro7886978635576167513Heap_a(F,X2),aa_Pro7886978635576167513Heap_a(F2,X2))
=> refine_Imp_refines_a(heap_T757603679106148408unit_a(M,F),heap_T757603679106148408unit_a(M4,F2)) ) ) ).
% refines_bind
tff(fact_227_refines__bind,axiom,
! [M: heap_T5738788834812785303t_unit,M4: heap_T5738788834812785303t_unit,F: fun_Pr292408699144966967t_unit,F2: fun_Pr292408699144966967t_unit] :
( refine451278484176762712t_unit(M,M4)
=> ( ! [X2: product_unit] : refine451278484176762712t_unit(aa_Pro1108561329362217960t_unit(F,X2),aa_Pro1108561329362217960t_unit(F2,X2))
=> refine451278484176762712t_unit(heap_T2633723481742716231t_unit(M,F),heap_T2633723481742716231t_unit(M4,F2)) ) ) ).
% refines_bind
tff(fact_228_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
% less_eq_nat.simps(1)
tff(fact_229_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
<=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
tff(fact_230_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
tff(fact_231_le__0__eq,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> ( N = zero_zero_nat ) ) ).
% le_0_eq
tff(fact_232_time__assert_H,axiom,
! [P: bool,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(refine_Imp_assert(P),H) = zero_zero_nat ).
% time_assert'
tff(fact_233_TBOUND__refines,axiom,
! [C: heap_Time_Heap_a,T: nat,D: heap_Time_Heap_a] :
( pp(aa_nat_bool(time_TBOUND_a(C),T))
=> ( refine_Imp_refines_a(C,D)
=> pp(aa_nat_bool(time_TBOUND_a(D),T)) ) ) ).
% TBOUND_refines
tff(fact_234_TBOUND__refines,axiom,
! [C: heap_T5738788834812785303t_unit,T: nat,D: heap_T5738788834812785303t_unit] :
( pp(aa_nat_bool(time_T7469515765551943773t_unit(C),T))
=> ( refine451278484176762712t_unit(C,D)
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(D),T)) ) ) ).
% TBOUND_refines
tff(fact_235_GreatestI__nat,axiom,
! [P: fun_nat_bool,K: nat,B: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).
% GreatestI_nat
tff(fact_236_Greatest__le__nat,axiom,
! [P: fun_nat_bool,K: nat,B: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).
% Greatest_le_nat
tff(fact_237_GreatestI__ex__nat,axiom,
! [P: fun_nat_bool,B: nat] :
( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).
% GreatestI_ex_nat
tff(fact_238_bind__ureturn,axiom,
! [F: heap_Time_Heap_a] : heap_Time_bind_a_a(F,heap_Time_ureturn_a) = F ).
% bind_ureturn
tff(fact_239_bind__ureturn,axiom,
! [F: heap_T5738788834812785303t_unit] : heap_T2633723481742716231t_unit(F,heap_T4284346855313245393t_unit) = F ).
% bind_ureturn
tff(fact_240_le__zero__eq,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> ( N = zero_zero_nat ) ) ).
% le_zero_eq
tff(fact_241_le__numeral__extra_I3_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).
% le_numeral_extra(3)
tff(fact_242_le__numeral__extra_I3_J,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),zero_z2226904508553997617atural)) ).
% le_numeral_extra(3)
tff(fact_243_le__numeral__extra_I3_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).
% le_numeral_extra(3)
tff(fact_244_le__numeral__extra_I3_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).
% le_numeral_extra(3)
tff(fact_245_le__numeral__extra_I3_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).
% le_numeral_extra(3)
tff(fact_246_zero__le,axiom,
! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).
% zero_le
tff(fact_247_time__refines,axiom,
! [C: heap_Time_Heap_a,C2: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
( refine_Imp_refines_a(C,C2)
=> ( ~ time_fails_a(C2,H)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_a(C,H)),time_time_a(C2,H))) ) ) ).
% time_refines
tff(fact_248_time__refines,axiom,
! [C: heap_T5738788834812785303t_unit,C2: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
( refine451278484176762712t_unit(C,C2)
=> ( ~ time_f8834461667527620124t_unit(C2,H)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t4224138285095624986t_unit(C,H)),time_t4224138285095624986t_unit(C2,H))) ) ) ).
% time_refines
tff(fact_249_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_250_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),zero_z2226904508553997617atural))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_251_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_252_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_253_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_254_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
tff(fact_255_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
<=> ( X = zero_zero_nat ) ) ).
% zero_reorient
tff(fact_256_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
<=> ( X = zero_zero_int ) ) ).
% zero_reorient
tff(fact_257_zero__reorient,axiom,
! [X: code_integer] :
( ( zero_z3403309356797280102nteger = X )
<=> ( X = zero_z3403309356797280102nteger ) ) ).
% zero_reorient
tff(fact_258_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
<=> ( X = zero_zero_rat ) ) ).
% zero_reorient
tff(fact_259_zero__reorient,axiom,
! [X: code_natural] :
( ( zero_z2226904508553997617atural = X )
<=> ( X = zero_z2226904508553997617atural ) ) ).
% zero_reorient
tff(fact_260_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_261_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_262_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_263_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_264_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_Code_natural(semiri3763490453095760265atural,M) = aa_nat_Code_natural(semiri3763490453095760265atural,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_265_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_266_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_267_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_268_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_269_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_Code_natural(semiri3763490453095760265atural,M) = zero_z2226904508553997617atural )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_270_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_271_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_272_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_273_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_274_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z2226904508553997617atural = aa_nat_Code_natural(semiri3763490453095760265atural,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_275_of__nat__0,axiom,
aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).
% of_nat_0
tff(fact_276_of__nat__0,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).
% of_nat_0
tff(fact_277_of__nat__0,axiom,
aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).
% of_nat_0
tff(fact_278_of__nat__0,axiom,
aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).
% of_nat_0
tff(fact_279_of__nat__0,axiom,
aa_nat_Code_natural(semiri3763490453095760265atural,zero_zero_nat) = zero_z2226904508553997617atural ).
% of_nat_0
tff(fact_280_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_281_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_282_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_283_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_284_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_285_int__ops_I1_J,axiom,
aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).
% int_ops(1)
tff(fact_286_nat__int__comparison_I3_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).
% nat_int_comparison(3)
tff(fact_287_nat__leq__as__int,axiom,
! [X3: nat,Xa: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).
% nat_leq_as_int
tff(fact_288_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).
% of_nat_0_le_iff
tff(fact_289_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(semiri3763490453095760265atural,N))) ).
% of_nat_0_le_iff
tff(fact_290_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).
% of_nat_0_le_iff
tff(fact_291_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).
% of_nat_0_le_iff
tff(fact_292_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% of_nat_0_le_iff
tff(fact_293_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J))) ) ).
% of_nat_mono
tff(fact_294_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,I)),aa_nat_Code_natural(semiri3763490453095760265atural,J))) ) ).
% of_nat_mono
tff(fact_295_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).
% of_nat_mono
tff(fact_296_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).
% of_nat_mono
tff(fact_297_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).
% of_nat_mono
tff(fact_298_fails__refines,axiom,
! [C: heap_T5738788834812785303t_unit,C2: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
( refine451278484176762712t_unit(C,C2)
=> ( time_f8834461667527620124t_unit(C,H)
=> time_f8834461667527620124t_unit(C2,H) ) ) ).
% fails_refines
tff(fact_299_fails__assert_H,axiom,
! [P: bool,H: heap_e7401611519738050253t_unit] :
( time_f8834461667527620124t_unit(refine_Imp_assert(P),H)
<=> ~ pp(P) ) ).
% fails_assert'
tff(fact_300_real__arch__simple,axiom,
! [X: rat] :
? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).
% real_arch_simple
tff(fact_301_time__wait,axiom,
! [N: nat,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(heap_Time_wait(N),H) = N ).
% time_wait
tff(fact_302_zle__int,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% zle_int
tff(fact_303_dbl__simps_I2_J,axiom,
neg_numeral_dbl_int(zero_zero_int) = zero_zero_int ).
% dbl_simps(2)
tff(fact_304_dbl__simps_I2_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% dbl_simps(2)
tff(fact_305_dbl__simps_I2_J,axiom,
aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).
% dbl_simps(2)
tff(fact_306_fails__bind,axiom,
! [M: heap_Time_Heap_a,F: fun_a_6093291656547882009Heap_a,H: heap_e7401611519738050253t_unit] :
( time_fails_a(heap_Time_bind_a_a(M,F),H)
<=> ( ~ time_fails_a(M,H)
=> time_fails_a(aa_a_H8829438773920479562Heap_a(F,time_the_res_a(M,H)),time_the_heap_a(M,H)) ) ) ).
% fails_bind
tff(fact_307_fails__bind,axiom,
! [M: heap_T5738788834812785303t_unit,F: fun_Pr5215307506330550056Heap_a,H: heap_e7401611519738050253t_unit] :
( time_fails_a(heap_T757603679106148408unit_a(M,F),H)
<=> ( ~ time_f8834461667527620124t_unit(M,H)
=> time_fails_a(aa_Pro7886978635576167513Heap_a(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H)) ) ) ).
% fails_bind
tff(fact_308_fails__bind,axiom,
! [M: heap_T5738788834812785303t_unit,F: fun_Pr292408699144966967t_unit,H: heap_e7401611519738050253t_unit] :
( time_f8834461667527620124t_unit(heap_T2633723481742716231t_unit(M,F),H)
<=> ( ~ time_f8834461667527620124t_unit(M,H)
=> time_f8834461667527620124t_unit(aa_Pro1108561329362217960t_unit(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H)) ) ) ).
% fails_bind
tff(fact_309_TBOUND__bind,axiom,
! [M: heap_Time_Heap_a,T_1: nat,F: fun_a_6093291656547882009Heap_a,T_2: nat] :
( pp(aa_nat_bool(time_TBOUND_a(M),T_1))
=> ( ! [X2: a,H2: heap_e7401611519738050253t_unit] :
( ( X2 = time_the_res_a(M,H2) )
=> ( ~ time_fails_a(M,H2)
=> pp(aa_nat_bool(time_TBOUND_a(aa_a_H8829438773920479562Heap_a(F,X2)),T_2)) ) )
=> pp(aa_nat_bool(time_TBOUND_a(heap_Time_bind_a_a(M,F)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,T_1),T_2))) ) ) ).
% TBOUND_bind
tff(fact_310_TBOUND__bind,axiom,
! [M: heap_Time_Heap_a,T_1: nat,F: fun_a_2990225006182978344t_unit,T_2: nat] :
( pp(aa_nat_bool(time_TBOUND_a(M),T_1))
=> ( ! [X2: a,H2: heap_e7401611519738050253t_unit] :
( ( X2 = time_the_res_a(M,H2) )
=> ( ~ time_fails_a(M,H2)
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(aa_a_H4862568901524997081t_unit(F,X2)),T_2)) ) )
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(heap_T8685611227969916822t_unit(M,F)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,T_1),T_2))) ) ) ).
% TBOUND_bind
tff(fact_311_TBOUND__bind,axiom,
! [M: heap_T5738788834812785303t_unit,T_1: nat,F: fun_Pr5215307506330550056Heap_a,T_2: nat] :
( pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T_1))
=> ( ! [X2: product_unit,H2: heap_e7401611519738050253t_unit] :
( ( X2 = time_t2745601220510131320t_unit(M,H2) )
=> ( ~ time_f8834461667527620124t_unit(M,H2)
=> pp(aa_nat_bool(time_TBOUND_a(aa_Pro7886978635576167513Heap_a(F,X2)),T_2)) ) )
=> pp(aa_nat_bool(time_TBOUND_a(heap_T757603679106148408unit_a(M,F)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,T_1),T_2))) ) ) ).
% TBOUND_bind
tff(fact_312_TBOUND__bind,axiom,
! [M: heap_T5738788834812785303t_unit,T_1: nat,F: fun_Pr292408699144966967t_unit,T_2: nat] :
( pp(aa_nat_bool(time_T7469515765551943773t_unit(M),T_1))
=> ( ! [X2: product_unit,H2: heap_e7401611519738050253t_unit] :
( ( X2 = time_t2745601220510131320t_unit(M,H2) )
=> ( ~ time_f8834461667527620124t_unit(M,H2)
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(aa_Pro1108561329362217960t_unit(F,X2)),T_2)) ) )
=> pp(aa_nat_bool(time_T7469515765551943773t_unit(heap_T2633723481742716231t_unit(M,F)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,T_1),T_2))) ) ) ).
% TBOUND_bind
tff(fact_313_add__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_314_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_315_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_316_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_317_add__right__cancel,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_318_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_319_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_320_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_321_add__le__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% add_le_cancel_left
tff(fact_322_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_cancel_left
tff(fact_323_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_cancel_left
tff(fact_324_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_cancel_left
tff(fact_325_add__le__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% add_le_cancel_right
tff(fact_326_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_cancel_right
tff(fact_327_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_cancel_right
tff(fact_328_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_cancel_right
tff(fact_329_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_330_double__eq__0__iff,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% double_eq_0_iff
tff(fact_331_double__eq__0__iff,axiom,
! [A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
tff(fact_332_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_333_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_334_add__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).
% add_0
tff(fact_335_add__0,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).
% add_0
tff(fact_336_add__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_z2226904508553997617atural),A) = A ).
% add_0
tff(fact_337_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_338_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_339_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_340_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_341_add__cancel__right__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_right
tff(fact_342_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
tff(fact_343_add__cancel__right__right,axiom,
! [A: code_natural,B: code_natural] :
( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B) )
<=> ( B = zero_z2226904508553997617atural ) ) ).
% add_cancel_right_right
tff(fact_344_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_345_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_346_add__cancel__right__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_left
tff(fact_347_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
tff(fact_348_add__cancel__right__left,axiom,
! [A: code_natural,B: code_natural] :
( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A) )
<=> ( B = zero_z2226904508553997617atural ) ) ).
% add_cancel_right_left
tff(fact_349_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_350_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_351_add__cancel__left__right,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = A )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_right
tff(fact_352_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
tff(fact_353_add__cancel__left__right,axiom,
! [A: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B) = A )
<=> ( B = zero_z2226904508553997617atural ) ) ).
% add_cancel_left_right
tff(fact_354_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_355_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_356_add__cancel__left__left,axiom,
! [B: code_integer,A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = A )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_left
tff(fact_357_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
tff(fact_358_add__cancel__left__left,axiom,
! [B: code_natural,A: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A) = A )
<=> ( B = zero_z2226904508553997617atural ) ) ).
% add_cancel_left_left
tff(fact_359_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_360_double__zero__sym,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% double_zero_sym
tff(fact_361_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
<=> ( A = zero_zero_rat ) ) ).
% double_zero_sym
tff(fact_362_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_363_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_364_add_Oright__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).
% add.right_neutral
tff(fact_365_add_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).
% add.right_neutral
tff(fact_366_add_Oright__neutral,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).
% add.right_neutral
tff(fact_367_add__is__0,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
tff(fact_368_Nat_Oadd__0__right,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).
% Nat.add_0_right
tff(fact_369_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% nat_add_left_cancel_le
tff(fact_370_add__le__same__cancel1,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% add_le_same_cancel1
tff(fact_371_add__le__same__cancel1,axiom,
! [B: code_natural,A: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)) ) ).
% add_le_same_cancel1
tff(fact_372_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% add_le_same_cancel1
tff(fact_373_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).
% add_le_same_cancel1
tff(fact_374_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% add_le_same_cancel1
tff(fact_375_add__le__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% add_le_same_cancel2
tff(fact_376_add__le__same__cancel2,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)) ) ).
% add_le_same_cancel2
tff(fact_377_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% add_le_same_cancel2
tff(fact_378_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).
% add_le_same_cancel2
tff(fact_379_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% add_le_same_cancel2
tff(fact_380_le__add__same__cancel1,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) ).
% le_add_same_cancel1
tff(fact_381_le__add__same__cancel1,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)) ) ).
% le_add_same_cancel1
tff(fact_382_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).
% le_add_same_cancel1
tff(fact_383_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).
% le_add_same_cancel1
tff(fact_384_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).
% le_add_same_cancel1
tff(fact_385_le__add__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) ).
% le_add_same_cancel2
tff(fact_386_le__add__same__cancel2,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)) ) ).
% le_add_same_cancel2
tff(fact_387_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).
% le_add_same_cancel2
tff(fact_388_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).
% le_add_same_cancel2
tff(fact_389_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).
% le_add_same_cancel2
tff(fact_390_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_391_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_392_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_393_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_394_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_395_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_396_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_add
tff(fact_397_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_add
tff(fact_398_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% of_nat_add
tff(fact_399_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% of_nat_add
tff(fact_400_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% of_nat_add
tff(fact_401_less__eq__int__code_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).
% less_eq_int_code(1)
tff(fact_402_nonneg__int__cases,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).
% nonneg_int_cases
tff(fact_403_zero__le__imp__eq__int,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).
% zero_le_imp_eq_int
tff(fact_404_int__if,axiom,
! [P: bool,A: nat,B: nat] :
( ( pp(P)
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,A) ) )
& ( ~ pp(P)
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ) ).
% int_if
tff(fact_405_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
tff(fact_406_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_407_is__num__normalize_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% is_num_normalize(1)
tff(fact_408_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% is_num_normalize(1)
tff(fact_409_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_410_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_411_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_412_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_413_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_414_group__cancel_Oadd1,axiom,
! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
( ( A3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A3),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% group_cancel.add1
tff(fact_415_group__cancel_Oadd1,axiom,
! [A3: rat,K: rat,A: rat,B: rat] :
( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A3),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% group_cancel.add1
tff(fact_416_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_417_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_418_group__cancel_Oadd2,axiom,
! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% group_cancel.add2
tff(fact_419_group__cancel_Oadd2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% group_cancel.add2
tff(fact_420_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_421_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_422_dbl__def,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X) ).
% dbl_def
tff(fact_423_dbl__def,axiom,
! [X: rat] : aa_rat_rat(neg_numeral_dbl_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X) ).
% dbl_def
tff(fact_424_dbl__def,axiom,
! [X: int] : neg_numeral_dbl_int(X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X) ).
% dbl_def
tff(fact_425_add_Oassoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% add.assoc
tff(fact_426_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% add.assoc
tff(fact_427_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_428_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_429_add_Oleft__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_430_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_431_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_432_add_Oright__cancel,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_433_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_434_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_435_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_436_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_437_ab__semigroup__add__class_Oadd_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) ).
% ab_semigroup_add_class.add.commute
tff(fact_438_ab__semigroup__add__class_Oadd_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) ).
% ab_semigroup_add_class.add.commute
tff(fact_439_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_440_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_441_ab__semigroup__add__class_Oadd_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)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_442_ab__semigroup__add__class_Oadd_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)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_443_add__left__imp__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_444_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_445_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_446_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_447_add__right__imp__eq,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_448_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_449_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_450_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_451_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( M = N ) ) ).
% int_int_eq
tff(fact_452_verit__la__generic,axiom,
! [A: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
| ( A = X )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A)) ) ).
% verit_la_generic
tff(fact_453_fails__wait,axiom,
! [X: nat,H: heap_e7401611519738050253t_unit] : ~ time_f8834461667527620124t_unit(heap_Time_wait(X),H) ).
% fails_wait
tff(fact_454_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_455_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( aa_list_int_nat(size_size_list_int,X) != aa_list_int_nat(size_size_list_int,Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_456_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_457_size__neq__size__imp__neq,axiom,
! [X: list_bool,Y: list_bool] :
( ( aa_list_bool_nat(size_size_list_bool,X) != aa_list_bool_nat(size_size_list_bool,Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_458_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( size_size_char(X) != size_size_char(Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_459_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),J))
& ( K = L ) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_460_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
& ( K = L ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_461_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& ( K = L ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_462_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
& ( K = L ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_463_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( ( I = J )
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_464_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_465_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_466_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_467_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),J))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_468_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_469_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_470_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_471_add__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D))) ) ) ).
% add_mono
tff(fact_472_add__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).
% add_mono
tff(fact_473_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_mono
tff(fact_474_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_mono
tff(fact_475_add__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))) ) ).
% add_left_mono
tff(fact_476_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).
% add_left_mono
tff(fact_477_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).
% add_left_mono
tff(fact_478_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% add_left_mono
tff(fact_479_less__eqE,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).
% less_eqE
tff(fact_480_add__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))) ) ).
% add_right_mono
tff(fact_481_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ).
% add_right_mono
tff(fact_482_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).
% add_right_mono
tff(fact_483_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).
% add_right_mono
tff(fact_484_le__iff__add,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).
% le_iff_add
tff(fact_485_add__le__imp__le__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% add_le_imp_le_left
tff(fact_486_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_imp_le_left
tff(fact_487_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_imp_le_left
tff(fact_488_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_imp_le_left
tff(fact_489_add__le__imp__le__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% add_le_imp_le_right
tff(fact_490_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_imp_le_right
tff(fact_491_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_imp_le_right
tff(fact_492_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_imp_le_right
tff(fact_493_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_494_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_495_verit__sum__simplify,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).
% verit_sum_simplify
tff(fact_496_verit__sum__simplify,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).
% verit_sum_simplify
tff(fact_497_verit__sum__simplify,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).
% verit_sum_simplify
tff(fact_498_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_499_add_Ogroup__left__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).
% add.group_left_neutral
tff(fact_500_add_Ogroup__left__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).
% add.group_left_neutral
tff(fact_501_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_502_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_503_add_Ocomm__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).
% add.comm_neutral
tff(fact_504_add_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).
% add.comm_neutral
tff(fact_505_add_Ocomm__neutral,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).
% add.comm_neutral
tff(fact_506_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_507_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_508_comm__monoid__add__class_Oadd__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_509_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_510_comm__monoid__add__class_Oadd__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_z2226904508553997617atural),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_511_plus__nat_Oadd__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).
% plus_nat.add_0
tff(fact_512_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
tff(fact_513_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).
% add_leE
tff(fact_514_le__add1,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).
% le_add1
tff(fact_515_le__add2,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
% le_add2
tff(fact_516_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% add_leD1
tff(fact_517_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).
% add_leD2
tff(fact_518_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
=> ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).
% le_Suc_ex
tff(fact_519_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).
% add_le_mono
tff(fact_520_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).
% add_le_mono1
tff(fact_521_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
% trans_le_add1
tff(fact_522_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
% trans_le_add2
tff(fact_523_nat__le__iff__add,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
<=> ? [K2: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).
% nat_le_iff_add
tff(fact_524_add__decreasing,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_525_add__decreasing,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,C),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_526_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_527_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_528_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_529_add__increasing,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C))) ) ) ).
% add_increasing
tff(fact_530_add__increasing,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C))) ) ) ).
% add_increasing
tff(fact_531_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).
% add_increasing
tff(fact_532_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_increasing
tff(fact_533_add__increasing,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_increasing
tff(fact_534_add__decreasing2,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_535_add__decreasing2,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,C),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_536_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_537_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_538_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_539_add__increasing2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C))) ) ) ).
% add_increasing2
tff(fact_540_add__increasing2,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C))) ) ) ).
% add_increasing2
tff(fact_541_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).
% add_increasing2
tff(fact_542_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_increasing2
tff(fact_543_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_increasing2
tff(fact_544_add__nonneg__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_545_add__nonneg__nonneg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_546_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_547_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_548_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_549_add__nonpos__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% add_nonpos_nonpos
tff(fact_550_add__nonpos__nonpos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% add_nonpos_nonpos
tff(fact_551_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).
% add_nonpos_nonpos
tff(fact_552_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_nonpos_nonpos
tff(fact_553_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_nonpos_nonpos
tff(fact_554_add__nonneg__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_555_add__nonneg__eq__0__iff,axiom,
! [X: code_natural,Y: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),X))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),Y))
=> ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Y) = zero_z2226904508553997617atural )
<=> ( ( X = zero_z2226904508553997617atural )
& ( Y = zero_z2226904508553997617atural ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_556_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_557_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_558_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_559_add__nonpos__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),zero_z3403309356797280102nteger))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_560_add__nonpos__eq__0__iff,axiom,
! [X: code_natural,Y: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,X),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,Y),zero_z2226904508553997617atural))
=> ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Y) = zero_z2226904508553997617atural )
<=> ( ( X = zero_z2226904508553997617atural )
& ( Y = zero_z2226904508553997617atural ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_561_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_562_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_563_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_564_time__bind,axiom,
! [M: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,F: fun_a_6093291656547882009Heap_a] :
( ( ( ~ time_fails_a(M,H)
& ~ time_fails_a(aa_a_H8829438773920479562Heap_a(F,time_the_res_a(M,H)),time_the_heap_a(M,H)) )
=> ( time_time_a(heap_Time_bind_a_a(M,F),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,time_time_a(M,H)),time_time_a(aa_a_H8829438773920479562Heap_a(F,time_the_res_a(M,H)),time_the_heap_a(M,H))) ) )
& ( ~ ( ~ time_fails_a(M,H)
& ~ time_fails_a(aa_a_H8829438773920479562Heap_a(F,time_the_res_a(M,H)),time_the_heap_a(M,H)) )
=> ( time_time_a(heap_Time_bind_a_a(M,F),H) = zero_zero_nat ) ) ) ).
% time_bind
tff(fact_565_time__bind,axiom,
! [M: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,F: fun_a_2990225006182978344t_unit] :
( ( ( ~ time_fails_a(M,H)
& ~ time_f8834461667527620124t_unit(aa_a_H4862568901524997081t_unit(F,time_the_res_a(M,H)),time_the_heap_a(M,H)) )
=> ( time_t4224138285095624986t_unit(heap_T8685611227969916822t_unit(M,F),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,time_time_a(M,H)),time_t4224138285095624986t_unit(aa_a_H4862568901524997081t_unit(F,time_the_res_a(M,H)),time_the_heap_a(M,H))) ) )
& ( ~ ( ~ time_fails_a(M,H)
& ~ time_f8834461667527620124t_unit(aa_a_H4862568901524997081t_unit(F,time_the_res_a(M,H)),time_the_heap_a(M,H)) )
=> ( time_t4224138285095624986t_unit(heap_T8685611227969916822t_unit(M,F),H) = zero_zero_nat ) ) ) ).
% time_bind
tff(fact_566_time__bind,axiom,
! [M: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,F: fun_Pr5215307506330550056Heap_a] :
( ( ( ~ time_f8834461667527620124t_unit(M,H)
& ~ time_fails_a(aa_Pro7886978635576167513Heap_a(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H)) )
=> ( time_time_a(heap_T757603679106148408unit_a(M,F),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,time_t4224138285095624986t_unit(M,H)),time_time_a(aa_Pro7886978635576167513Heap_a(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H))) ) )
& ( ~ ( ~ time_f8834461667527620124t_unit(M,H)
& ~ time_fails_a(aa_Pro7886978635576167513Heap_a(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H)) )
=> ( time_time_a(heap_T757603679106148408unit_a(M,F),H) = zero_zero_nat ) ) ) ).
% time_bind
tff(fact_567_time__bind,axiom,
! [M: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,F: fun_Pr292408699144966967t_unit] :
( ( ( ~ time_f8834461667527620124t_unit(M,H)
& ~ time_f8834461667527620124t_unit(aa_Pro1108561329362217960t_unit(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H)) )
=> ( time_t4224138285095624986t_unit(heap_T2633723481742716231t_unit(M,F),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,time_t4224138285095624986t_unit(M,H)),time_t4224138285095624986t_unit(aa_Pro1108561329362217960t_unit(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H))) ) )
& ( ~ ( ~ time_f8834461667527620124t_unit(M,H)
& ~ time_f8834461667527620124t_unit(aa_Pro1108561329362217960t_unit(F,time_t2745601220510131320t_unit(M,H)),time_t3350536682685678164t_unit(M,H)) )
=> ( time_t4224138285095624986t_unit(heap_T2633723481742716231t_unit(M,F),H) = zero_zero_nat ) ) ) ).
% time_bind
tff(fact_568_imp__le__cong,axiom,
! [X: int,X5: int,P: bool,P2: bool] :
( ( X = X5 )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
=> ( pp(P)
<=> pp(P2) ) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> pp(P) )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
=> pp(P2) ) ) ) ) ).
% imp_le_cong
tff(fact_569_conj__le__cong,axiom,
! [X: int,X5: int,P: bool,P2: bool] :
( ( X = X5 )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
=> ( pp(P)
<=> pp(P2) ) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
& pp(P) )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
& pp(P2) ) ) ) ) ).
% conj_le_cong
tff(fact_570_Euclid__induct,axiom,
! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B2),A2)) )
=> ( ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),zero_zero_nat))
=> ( ! [A2: nat,B2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).
% Euclid_induct
tff(fact_571_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_572_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_573_add__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% add_0_iff
tff(fact_574_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
<=> ( A = zero_zero_rat ) ) ).
% add_0_iff
tff(fact_575_of__nat__code,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = semiri8420488043553186161ux_int(aTP_Lamm_ao,N,zero_zero_int) ).
% of_nat_code
tff(fact_576_of__nat__code,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_ap),N),zero_zero_nat) ).
% of_nat_code
tff(fact_577_of__nat__code,axiom,
! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,N) = aa_Cod1783974652533396728nteger(semiri4055485073559036834nteger(aTP_Lamm_aq,N),zero_z3403309356797280102nteger) ).
% of_nat_code
tff(fact_578_of__nat__code,axiom,
! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_ar,N),zero_zero_rat) ).
% of_nat_code
tff(fact_579_of__nat__code,axiom,
! [N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,N) = semiri2879080225315754349atural(aTP_Lamm_as,N,zero_z2226904508553997617atural) ).
% of_nat_code
tff(fact_580_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_581_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_582_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_583_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_584_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_Code_natural(semiri3763490453095760265atural,N) = one_one_Code_natural )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_585_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_586_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_587_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_588_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_589_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_natural = aa_nat_Code_natural(semiri3763490453095760265atural,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_590_of__nat__1,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% of_nat_1
tff(fact_591_of__nat__1,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).
% of_nat_1
tff(fact_592_of__nat__1,axiom,
aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).
% of_nat_1
tff(fact_593_of__nat__1,axiom,
aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).
% of_nat_1
tff(fact_594_of__nat__1,axiom,
aa_nat_Code_natural(semiri3763490453095760265atural,one_one_nat) = one_one_Code_natural ).
% of_nat_1
tff(fact_595_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
<=> ( X = one_one_int ) ) ).
% one_reorient
tff(fact_596_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
<=> ( X = one_one_nat ) ) ).
% one_reorient
tff(fact_597_one__reorient,axiom,
! [X: code_integer] :
( ( one_one_Code_integer = X )
<=> ( X = one_one_Code_integer ) ) ).
% one_reorient
tff(fact_598_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
<=> ( X = one_one_rat ) ) ).
% one_reorient
tff(fact_599_one__reorient,axiom,
! [X: code_natural] :
( ( one_one_Code_natural = X )
<=> ( X = one_one_Code_natural ) ) ).
% one_reorient
tff(fact_600_odd__nonzero,axiom,
! [Z: 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 ).
% odd_nonzero
tff(fact_601_int__ge__induct,axiom,
! [K: int,I: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
=> ( pp(aa_int_bool(P,K))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_ge_induct
tff(fact_602_int__ops_I2_J,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% int_ops(2)
tff(fact_603_plus__int__code_I2_J,axiom,
! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).
% plus_int_code(2)
tff(fact_604_plus__int__code_I1_J,axiom,
! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).
% plus_int_code(1)
tff(fact_605_le__numeral__extra_I4_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),one_one_Code_integer)) ).
% le_numeral_extra(4)
tff(fact_606_le__numeral__extra_I4_J,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),one_one_Code_natural)) ).
% le_numeral_extra(4)
tff(fact_607_le__numeral__extra_I4_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).
% le_numeral_extra(4)
tff(fact_608_le__numeral__extra_I4_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).
% le_numeral_extra(4)
tff(fact_609_le__numeral__extra_I4_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).
% le_numeral_extra(4)
tff(fact_610_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).
% zadd_int_left
tff(fact_611_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_612_int__plus,axiom,
! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).
% int_plus
tff(fact_613_zle__iff__zadd,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z))
<=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).
% zle_iff_zadd
tff(fact_614_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),X))
<=> ( X != zero_zero_nat ) ) ).
% nat_geq_1_eq_neqz
tff(fact_615_someI2__bex,axiom,
! [A3: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
( ? [X3: product_prod_nat_nat] :
( member8440522571783428010at_nat(X3,A3)
& pp(aa_Pro2232559814647311250t_bool(P,X3)) )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat(X2,A3)
& pp(aa_Pro2232559814647311250t_bool(P,X2)) )
=> pp(aa_Pro2232559814647311250t_bool(Q,X2)) )
=> pp(aa_Pro2232559814647311250t_bool(Q,fChoic6978938873391328853at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_at(A3),P)))) ) ) ).
% someI2_bex
tff(fact_616_someI2__bex,axiom,
! [A3: set_se6260736226359567993nt_int,P: fun_se3032438776331272065t_bool,Q: fun_se3032438776331272065t_bool] :
( ? [X3: set_Pr958786334691620121nt_int] :
( member2340774599025711042nt_int(X3,A3)
& pp(aa_set7317184552576006522t_bool(P,X3)) )
=> ( ! [X2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int(X2,A3)
& pp(aa_set7317184552576006522t_bool(P,X2)) )
=> pp(aa_set7317184552576006522t_bool(Q,X2)) )
=> pp(aa_set7317184552576006522t_bool(Q,fChoic2749331249758569069nt_int(aa_fun4116018846136612672t_bool(aTP_Lamm_au(A3),P)))) ) ) ).
% someI2_bex
tff(fact_617_someI2__bex,axiom,
! [A3: set_set_nat,P: fun_set_nat_bool,Q: fun_set_nat_bool] :
( ? [X3: set_nat] :
( member_set_nat(X3,A3)
& pp(aa_set_nat_bool(P,X3)) )
=> ( ! [X2: set_nat] :
( ( member_set_nat(X2,A3)
& pp(aa_set_nat_bool(P,X2)) )
=> pp(aa_set_nat_bool(Q,X2)) )
=> pp(aa_set_nat_bool(Q,fChoice_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_av(A3),P)))) ) ) ).
% someI2_bex
tff(fact_618_someI2__bex,axiom,
! [A3: set_nat,P: fun_nat_bool,Q: fun_nat_bool] :
( ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A3))
& pp(aa_nat_bool(P,X3)) )
=> ( ! [X2: nat] :
( ( pp(aa_set_nat_bool(member_nat(X2),A3))
& pp(aa_nat_bool(P,X2)) )
=> pp(aa_nat_bool(Q,X2)) )
=> pp(aa_nat_bool(Q,fChoice_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(A3),P)))) ) ) ).
% someI2_bex
tff(fact_619_someI2__bex,axiom,
! [A3: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ? [X3: int] :
( member_int(X3,A3)
& pp(aa_int_bool(P,X3)) )
=> ( ! [X2: int] :
( ( member_int(X2,A3)
& pp(aa_int_bool(P,X2)) )
=> pp(aa_int_bool(Q,X2)) )
=> pp(aa_int_bool(Q,fChoice_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ax(A3),P)))) ) ) ).
% someI2_bex
tff(fact_620_not__one__le__zero,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).
% not_one_le_zero
tff(fact_621_not__one__le__zero,axiom,
~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),zero_z2226904508553997617atural)) ).
% not_one_le_zero
tff(fact_622_not__one__le__zero,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).
% not_one_le_zero
tff(fact_623_not__one__le__zero,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
% not_one_le_zero
tff(fact_624_not__one__le__zero,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
% not_one_le_zero
tff(fact_625_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_626_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),one_one_Code_natural)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_627_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_628_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_629_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_630_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).
% zero_less_one_class.zero_le_one
tff(fact_631_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),one_one_Code_natural)) ).
% zero_less_one_class.zero_le_one
tff(fact_632_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).
% zero_less_one_class.zero_le_one
tff(fact_633_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
% zero_less_one_class.zero_le_one
tff(fact_634_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
% zero_less_one_class.zero_le_one
tff(fact_635_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
tff(fact_636_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
tff(fact_637_zero__neq__one,axiom,
zero_z3403309356797280102nteger != one_one_Code_integer ).
% zero_neq_one
tff(fact_638_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
tff(fact_639_zero__neq__one,axiom,
zero_z2226904508553997617atural != one_one_Code_natural ).
% zero_neq_one
tff(fact_640_return__bind,axiom,
! [X: product_unit,F: fun_Pr5215307506330550056Heap_a] : heap_T757603679106148408unit_a(aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X),F) = heap_T757603679106148408unit_a(heap_Time_wait(one_one_nat),aa_fun8407955803497444028Heap_a(aTP_Lamm_ay(X),F)) ).
% return_bind
tff(fact_641_return__bind,axiom,
! [X: product_unit,F: fun_Pr292408699144966967t_unit] : heap_T2633723481742716231t_unit(aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X),F) = heap_T2633723481742716231t_unit(heap_Time_wait(one_one_nat),aa_fun7583176353580103804t_unit(aTP_Lamm_az(X),F)) ).
% return_bind
tff(fact_642_return__bind,axiom,
! [X: a,F: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,X),F) = heap_T757603679106148408unit_a(heap_Time_wait(one_one_nat),aa_fun7182454221404266251Heap_a(aTP_Lamm_ba(X),F)) ).
% return_bind
tff(fact_643_bind__return,axiom,
! [F: heap_T5738788834812785303t_unit] : heap_T2633723481742716231t_unit(F,heap_T7507251653302230130t_unit) = heap_T2633723481742716231t_unit(heap_Time_wait(one_one_nat),aTP_Lamm_an(F)) ).
% bind_return
tff(fact_644_bind__return,axiom,
! [F: heap_Time_Heap_a] : heap_Time_bind_a_a(F,heap_Time_return_a) = heap_T757603679106148408unit_a(heap_Time_wait(one_one_nat),aTP_Lamm_am(F)) ).
% bind_return
tff(fact_645_time__array__nth,axiom,
! [P3: array_Product_unit,I: nat,H: heap_e7401611519738050253t_unit] :
( ( time_f8834461667527620124t_unit(array_7872002506669749220t_unit(P3,I),H)
=> ( time_t4224138285095624986t_unit(array_7872002506669749220t_unit(P3,I),H) = zero_zero_nat ) )
& ( ~ time_f8834461667527620124t_unit(array_7872002506669749220t_unit(P3,I),H)
=> ( time_t4224138285095624986t_unit(array_7872002506669749220t_unit(P3,I),H) = one_one_nat ) ) ) ).
% time_array_nth
tff(fact_646_dbl__inc__simps_I2_J,axiom,
neg_nu5851722552734809277nc_int(zero_zero_int) = one_one_int ).
% dbl_inc_simps(2)
tff(fact_647_dbl__inc__simps_I2_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,zero_z3403309356797280102nteger) = one_one_Code_integer ).
% dbl_inc_simps(2)
tff(fact_648_dbl__inc__simps_I2_J,axiom,
aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).
% dbl_inc_simps(2)
tff(fact_649_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
tff(fact_650_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
tff(fact_651_time__return,axiom,
! [X: a,H: heap_e7401611519738050253t_unit] : time_time_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,X),H) = one_one_nat ).
% time_return
tff(fact_652_time__return,axiom,
! [X: product_unit,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X),H) = one_one_nat ).
% time_return
tff(fact_653_fails__return,axiom,
! [X: product_unit,H: heap_e7401611519738050253t_unit] : ~ time_f8834461667527620124t_unit(aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X),H) ).
% fails_return
tff(fact_654_dbl__inc__def,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).
% dbl_inc_def
tff(fact_655_dbl__inc__def,axiom,
! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).
% dbl_inc_def
tff(fact_656_dbl__inc__def,axiom,
! [X: int] : neg_nu5851722552734809277nc_int(X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).
% dbl_inc_def
tff(fact_657_bind__lift,axiom,
! [F: heap_T5738788834812785303t_unit,G: fun_Product_unit_a] : heap_T757603679106148408unit_a(F,heap_T6901581777565460678unit_a(G)) = heap_T757603679106148408unit_a(F,aTP_Lamm_bb(G)) ).
% bind_lift
tff(fact_658_bind__lift,axiom,
! [F: heap_T5738788834812785303t_unit,G: fun_Pr2191293291710260995t_unit] : heap_T2633723481742716231t_unit(F,heap_T7011250591035762645t_unit(G)) = heap_T2633723481742716231t_unit(F,aTP_Lamm_bc(G)) ).
% bind_lift
tff(fact_659_bind__lift,axiom,
! [F: heap_Time_Heap_a,G: fun_a_a] : heap_Time_bind_a_a(F,heap_Time_lift_a_a(G)) = heap_Time_bind_a_a(F,aTP_Lamm_bd(G)) ).
% bind_lift
tff(fact_660_lookup__chain,axiom,
! [R: ref_Product_unit,F: heap_Time_Heap_a] : heap_T757603679106148408unit_a(ref_lo7930775386976318366t_unit(R),aTP_Lamm_am(F)) = heap_T757603679106148408unit_a(heap_Time_wait(one_one_nat),aTP_Lamm_am(F)) ).
% lookup_chain
tff(fact_661_lookup__chain,axiom,
! [R: ref_Product_unit,F: heap_T5738788834812785303t_unit] : heap_T2633723481742716231t_unit(ref_lo7930775386976318366t_unit(R),aTP_Lamm_an(F)) = heap_T2633723481742716231t_unit(heap_Time_wait(one_one_nat),aTP_Lamm_an(F)) ).
% lookup_chain
tff(fact_662_ceiling__add__one,axiom,
! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).
% ceiling_add_one
tff(fact_663_ceiling__le__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).
% ceiling_le_one
tff(fact_664_ceiling__le__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).
% ceiling_le_zero
tff(fact_665_of__int__le__1__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).
% of_int_le_1_iff
tff(fact_666_of__int__le__1__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).
% of_int_le_1_iff
tff(fact_667_of__int__le__1__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).
% of_int_le_1_iff
tff(fact_668_of__int__1__le__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).
% of_int_1_le_iff
tff(fact_669_of__int__1__le__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).
% of_int_1_le_iff
tff(fact_670_of__int__1__le__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).
% of_int_1_le_iff
tff(fact_671_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,W) = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
<=> ( W = Z ) ) ).
% of_int_eq_iff
tff(fact_672_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( aa_int_rat(ring_1_of_int_rat,W) = aa_int_rat(ring_1_of_int_rat,Z) )
<=> ( W = Z ) ) ).
% of_int_eq_iff
tff(fact_673_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( aa_int_int(ring_1_of_int_int,W) = aa_int_int(ring_1_of_int_int,Z) )
<=> ( W = Z ) ) ).
% of_int_eq_iff
tff(fact_674_ceiling__of__int,axiom,
! [Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_int_rat(ring_1_of_int_rat,Z)) = Z ).
% ceiling_of_int
tff(fact_675_of__int__0,axiom,
aa_int_Code_integer(ring_18347121197199848620nteger,zero_zero_int) = zero_z3403309356797280102nteger ).
% of_int_0
tff(fact_676_of__int__0,axiom,
aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).
% of_int_0
tff(fact_677_of__int__0,axiom,
aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).
% of_int_0
tff(fact_678_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_z3403309356797280102nteger = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
<=> ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
tff(fact_679_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
<=> ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
tff(fact_680_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
<=> ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
tff(fact_681_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = zero_z3403309356797280102nteger )
<=> ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
tff(fact_682_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
<=> ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
tff(fact_683_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
<=> ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
tff(fact_684_of__int__le__iff,axiom,
! [W: int,Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% of_int_le_iff
tff(fact_685_of__int__le__iff,axiom,
! [W: int,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% of_int_le_iff
tff(fact_686_of__int__le__iff,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% of_int_le_iff
tff(fact_687_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = one_one_Code_integer )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_688_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_689_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_690_of__int__1,axiom,
aa_int_Code_integer(ring_18347121197199848620nteger,one_one_int) = one_one_Code_integer ).
% of_int_1
tff(fact_691_of__int__1,axiom,
aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).
% of_int_1
tff(fact_692_of__int__1,axiom,
aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).
% of_int_1
tff(fact_693_of__int__add,axiom,
! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).
% of_int_add
tff(fact_694_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(aa_rat_fun_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_695_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_696_ceiling__zero,axiom,
aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).
% ceiling_zero
tff(fact_697_of__int__of__nat__eq,axiom,
! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% of_int_of_nat_eq
tff(fact_698_of__int__of__nat__eq,axiom,
! [N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).
% of_int_of_nat_eq
tff(fact_699_of__int__of__nat__eq,axiom,
! [N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).
% of_int_of_nat_eq
tff(fact_700_ceiling__one,axiom,
aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).
% ceiling_one
tff(fact_701_ceiling__of__nat,axiom,
! [N: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% ceiling_of_nat
tff(fact_702_ceiling__add__of__int,axiom,
! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z) ).
% ceiling_add_of_int
tff(fact_703_of__int__le__0__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).
% of_int_le_0_iff
tff(fact_704_of__int__le__0__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).
% of_int_le_0_iff
tff(fact_705_of__int__le__0__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).
% of_int_le_0_iff
tff(fact_706_of__int__0__le__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).
% of_int_0_le_iff
tff(fact_707_of__int__0__le__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).
% of_int_0_le_iff
tff(fact_708_of__int__0__le__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).
% of_int_0_le_iff
tff(fact_709_le__of__int__ceiling,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ).
% le_of_int_ceiling
tff(fact_710_ceiling__le__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% ceiling_le_iff
tff(fact_711_ex__le__of__int,axiom,
! [X: rat] :
? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).
% ex_le_of_int
tff(fact_712_ceiling__mono,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).
% ceiling_mono
tff(fact_713_ceiling__add__le,axiom,
! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))) ).
% ceiling_add_le
tff(fact_714_of__int__nonneg,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))) ) ).
% of_int_nonneg
tff(fact_715_of__int__nonneg,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% of_int_nonneg
tff(fact_716_of__int__nonneg,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).
% of_int_nonneg
tff(fact_717_change__def,axiom,
! [F: fun_Pr2191293291710260995t_unit,R: ref_Product_unit] : ref_ch7259622376331601608t_unit(F,R) = heap_T2633723481742716231t_unit(ref_lo7930775386976318366t_unit(R),aa_ref3209944945879839949t_unit(aTP_Lamm_bf(F),R)) ).
% change_def
tff(fact_718_one__le__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).
% one_le_ceiling
tff(fact_719_ceiling__less__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).
% ceiling_less_one
tff(fact_720_of__nat__nat,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).
% of_nat_nat
tff(fact_721_of__nat__nat,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,Z)) = aa_int_Code_integer(ring_18347121197199848620nteger,Z) ) ) ).
% of_nat_nat
tff(fact_722_of__nat__nat,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).
% of_nat_nat
tff(fact_723_one__le__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).
% one_le_floor
tff(fact_724_power__int__decreasing,axiom,
! [N: int,N4: int,A: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> ( ( ( A != zero_zero_rat )
| ( N4 != zero_zero_int )
| ( N = zero_zero_int ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(A),N4)),aa_int_rat(power_int_rat(A),N))) ) ) ) ) ).
% power_int_decreasing
tff(fact_725_power__int__le__one,axiom,
! [X: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(X),N)),one_one_rat)) ) ) ) ).
% power_int_le_one
tff(fact_726_zero__le__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X)) ) ).
% zero_le_floor
tff(fact_727_rotate1__length01,axiom,
! [Xs: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
=> ( rotate1_nat(Xs) = Xs ) ) ).
% rotate1_length01
tff(fact_728_rotate1__length01,axiom,
! [Xs: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat))
=> ( rotate1_int(Xs) = Xs ) ) ).
% rotate1_length01
tff(fact_729_rotate1__length01,axiom,
! [Xs: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat))
=> ( rotate1_bool(Xs) = Xs ) ) ).
% rotate1_length01
tff(fact_730_le__imp__0__less,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).
% le_imp_0_less
tff(fact_731_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).
% bot_nat_0.not_eq_extremum
tff(fact_732_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% neq0_conv
tff(fact_733_less__nat__zero__code,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% less_nat_zero_code
tff(fact_734_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% nat_add_left_cancel_less
tff(fact_735_not__gr__zero,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ( N = zero_zero_nat ) ) ).
% not_gr_zero
tff(fact_736_add__less__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% add_less_cancel_left
tff(fact_737_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_cancel_left
tff(fact_738_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_cancel_left
tff(fact_739_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_cancel_left
tff(fact_740_add__less__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% add_less_cancel_right
tff(fact_741_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_cancel_right
tff(fact_742_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_cancel_right
tff(fact_743_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_cancel_right
tff(fact_744_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_745_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_746_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_747_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_748_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_749_add__gr__0,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% add_gr_0
tff(fact_750_less__one,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
<=> ( N = zero_zero_nat ) ) ).
% less_one
tff(fact_751_power__int__1__left,axiom,
! [N: int] : aa_int_rat(power_int_rat(one_one_rat),N) = one_one_rat ).
% power_int_1_left
tff(fact_752_nat__int,axiom,
! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).
% nat_int
tff(fact_753_floor__of__int,axiom,
! [Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_int_rat(ring_1_of_int_rat,Z)) = Z ).
% floor_of_int
tff(fact_754_length__rotate1,axiom,
! [Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,rotate1_nat(Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).
% length_rotate1
tff(fact_755_length__rotate1,axiom,
! [Xs: list_int] : aa_list_int_nat(size_size_list_int,rotate1_int(Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).
% length_rotate1
tff(fact_756_length__rotate1,axiom,
! [Xs: list_bool] : aa_list_bool_nat(size_size_list_bool,rotate1_bool(Xs)) = aa_list_bool_nat(size_size_list_bool,Xs) ).
% length_rotate1
tff(fact_757_add__less__same__cancel1,axiom,
! [B: code_natural,A: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)) ) ).
% add_less_same_cancel1
tff(fact_758_add__less__same__cancel1,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% add_less_same_cancel1
tff(fact_759_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% add_less_same_cancel1
tff(fact_760_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).
% add_less_same_cancel1
tff(fact_761_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% add_less_same_cancel1
tff(fact_762_add__less__same__cancel2,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)) ) ).
% add_less_same_cancel2
tff(fact_763_add__less__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% add_less_same_cancel2
tff(fact_764_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% add_less_same_cancel2
tff(fact_765_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).
% add_less_same_cancel2
tff(fact_766_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% add_less_same_cancel2
tff(fact_767_less__add__same__cancel1,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)) ) ).
% less_add_same_cancel1
tff(fact_768_less__add__same__cancel1,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ).
% less_add_same_cancel1
tff(fact_769_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).
% less_add_same_cancel1
tff(fact_770_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).
% less_add_same_cancel1
tff(fact_771_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).
% less_add_same_cancel1
tff(fact_772_less__add__same__cancel2,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)) ) ).
% less_add_same_cancel2
tff(fact_773_less__add__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ).
% less_add_same_cancel2
tff(fact_774_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).
% less_add_same_cancel2
tff(fact_775_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).
% less_add_same_cancel2
tff(fact_776_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).
% less_add_same_cancel2
tff(fact_777_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_778_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_779_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_780_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_781_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_782_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_783_of__int__less__iff,axiom,
! [W: int,Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% of_int_less_iff
tff(fact_784_of__int__less__iff,axiom,
! [W: int,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% of_int_less_iff
tff(fact_785_of__int__less__iff,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% of_int_less_iff
tff(fact_786_power__int__eq__0__iff,axiom,
! [X: rat,N: int] :
( ( aa_int_rat(power_int_rat(X),N) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( N != zero_zero_int ) ) ) ).
% power_int_eq_0_iff
tff(fact_787_power__int__0__left,axiom,
! [M: int] :
( ( M != zero_zero_int )
=> ( aa_int_rat(power_int_rat(zero_zero_rat),M) = zero_zero_rat ) ) ).
% power_int_0_left
tff(fact_788_floor__zero,axiom,
aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).
% floor_zero
tff(fact_789_power__int__0__right,axiom,
! [X: rat] : aa_int_rat(power_int_rat(X),zero_zero_int) = one_one_rat ).
% power_int_0_right
tff(fact_790_zless__nat__conj,axiom,
! [W: int,Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).
% zless_nat_conj
tff(fact_791_floor__one,axiom,
aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).
% floor_one
tff(fact_792_floor__of__nat,axiom,
! [N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% floor_of_nat
tff(fact_793_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_794_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_795_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_796_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_797_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_nat_Code_natural(semiri3763490453095760265atural,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_798_nat__0__iff,axiom,
! [I: int] :
( ( aa_int_nat(nat2,I) = zero_zero_nat )
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).
% nat_0_iff
tff(fact_799_nat__le__0,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
=> ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).
% nat_le_0
tff(fact_800_zero__less__nat__eq,axiom,
! [Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% zero_less_nat_eq
tff(fact_801_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% zle_add1_eq_le
tff(fact_802_int__nat__eq,axiom,
! [Z: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).
% int_nat_eq
tff(fact_803_of__int__less__0__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% of_int_less_0_iff
tff(fact_804_of__int__less__0__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% of_int_less_0_iff
tff(fact_805_of__int__less__0__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% of_int_less_0_iff
tff(fact_806_of__int__0__less__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% of_int_0_less_iff
tff(fact_807_of__int__0__less__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% of_int_0_less_iff
tff(fact_808_of__int__0__less__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% of_int_0_less_iff
tff(fact_809_power__int__mono__iff,axiom,
! [A: rat,B: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(B),N)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).
% power_int_mono_iff
tff(fact_810_of__int__less__1__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).
% of_int_less_1_iff
tff(fact_811_of__int__less__1__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).
% of_int_less_1_iff
tff(fact_812_of__int__less__1__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).
% of_int_less_1_iff
tff(fact_813_of__int__1__less__iff,axiom,
! [Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% of_int_1_less_iff
tff(fact_814_of__int__1__less__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% of_int_1_less_iff
tff(fact_815_of__int__1__less__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% of_int_1_less_iff
tff(fact_816_floor__less__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat)) ) ).
% floor_less_zero
tff(fact_817_zero__less__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).
% zero_less_floor
tff(fact_818_floor__le__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).
% floor_le_zero
tff(fact_819_zero__less__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).
% zero_less_ceiling
tff(fact_820_floor__less__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).
% floor_less_one
tff(fact_821_one__less__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).
% one_less_ceiling
tff(fact_822_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) != aa_list_nat_nat(size_size_list_nat,Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_823_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( aa_list_int_nat(size_size_list_int,Xs) != aa_list_int_nat(size_size_list_int,Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_824_neq__if__length__neq,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( aa_list_bool_nat(size_size_list_bool,Xs) != aa_list_bool_nat(size_size_list_bool,Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_825_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] : aa_list_nat_nat(size_size_list_nat,Xs2) = N ).
% Ex_list_of_length
tff(fact_826_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] : aa_list_int_nat(size_size_list_int,Xs2) = N ).
% Ex_list_of_length
tff(fact_827_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_bool] : aa_list_bool_nat(size_size_list_bool,Xs2) = N ).
% Ex_list_of_length
tff(fact_828_length__induct,axiom,
! [P: fun_list_nat_bool,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys2)),aa_list_nat_nat(size_size_list_nat,Xs2)))
=> pp(aa_list_nat_bool(P,Ys2)) )
=> pp(aa_list_nat_bool(P,Xs2)) )
=> pp(aa_list_nat_bool(P,Xs)) ) ).
% length_induct
tff(fact_829_length__induct,axiom,
! [P: fun_list_int_bool,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_int_nat(size_size_list_int,Ys2)),aa_list_int_nat(size_size_list_int,Xs2)))
=> pp(aa_list_int_bool(P,Ys2)) )
=> pp(aa_list_int_bool(P,Xs2)) )
=> pp(aa_list_int_bool(P,Xs)) ) ).
% length_induct
tff(fact_830_length__induct,axiom,
! [P: fun_list_bool_bool,Xs: list_bool] :
( ! [Xs2: list_bool] :
( ! [Ys2: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,Ys2)),aa_list_bool_nat(size_size_list_bool,Xs2)))
=> pp(aa_list_bool_bool(P,Ys2)) )
=> pp(aa_list_bool_bool(P,Xs2)) )
=> pp(aa_list_bool_bool(P,Xs)) ) ).
% length_induct
tff(fact_831_floor__less__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% floor_less_iff
tff(fact_832_minf_I7_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,T),X3)) ) ).
% minf(7)
tff(fact_833_minf_I7_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X3)) ) ).
% minf(7)
tff(fact_834_minf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X3)) ) ).
% minf(7)
tff(fact_835_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X3)) ) ).
% minf(7)
tff(fact_836_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3)) ) ).
% minf(7)
tff(fact_837_minf_I5_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),T)) ) ).
% minf(5)
tff(fact_838_minf_I5_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),T)) ) ).
% minf(5)
tff(fact_839_minf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),T)) ) ).
% minf(5)
tff(fact_840_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),T)) ) ).
% minf(5)
tff(fact_841_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T)) ) ).
% minf(5)
tff(fact_842_minf_I4_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> ( X3 != T ) ) ).
% minf(4)
tff(fact_843_minf_I4_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( X3 != T ) ) ).
% minf(4)
tff(fact_844_minf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( X3 != T ) ) ).
% minf(4)
tff(fact_845_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( X3 != T ) ) ).
% minf(4)
tff(fact_846_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( X3 != T ) ) ).
% minf(4)
tff(fact_847_minf_I3_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> ( X3 != T ) ) ).
% minf(3)
tff(fact_848_minf_I3_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( X3 != T ) ) ).
% minf(3)
tff(fact_849_minf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( X3 != T ) ) ).
% minf(3)
tff(fact_850_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( X3 != T ) ) ).
% minf(3)
tff(fact_851_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( X3 != T ) ) ).
% minf(3)
tff(fact_852_minf_I2_J,axiom,
! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q3: fun_Co8822979462182099027r_bool] :
( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z3))
=> ( pp(aa_Code_integer_bool(P,X2))
<=> pp(aa_Code_integer_bool(P2,X2)) ) )
=> ( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z3))
=> ( pp(aa_Code_integer_bool(Q,X2))
<=> pp(aa_Code_integer_bool(Q3,X2)) ) )
=> ? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> ( ( pp(aa_Code_integer_bool(P,X3))
| pp(aa_Code_integer_bool(Q,X3)) )
<=> ( pp(aa_Code_integer_bool(P2,X3))
| pp(aa_Code_integer_bool(Q3,X3)) ) ) ) ) ) ).
% minf(2)
tff(fact_853_minf_I2_J,axiom,
! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
=> ( pp(aa_rat_bool(P,X2))
<=> pp(aa_rat_bool(P2,X2)) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
=> ( pp(aa_rat_bool(Q,X2))
<=> pp(aa_rat_bool(Q3,X2)) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( ( pp(aa_rat_bool(P,X3))
| pp(aa_rat_bool(Q,X3)) )
<=> ( pp(aa_rat_bool(P2,X3))
| pp(aa_rat_bool(Q3,X3)) ) ) ) ) ) ).
% minf(2)
tff(fact_854_minf_I2_J,axiom,
! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
=> ( pp(aa_num_bool(P,X2))
<=> pp(aa_num_bool(P2,X2)) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
=> ( pp(aa_num_bool(Q,X2))
<=> pp(aa_num_bool(Q3,X2)) ) )
=> ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( ( pp(aa_num_bool(P,X3))
| pp(aa_num_bool(Q,X3)) )
<=> ( pp(aa_num_bool(P2,X3))
| pp(aa_num_bool(Q3,X3)) ) ) ) ) ) ).
% minf(2)
tff(fact_855_minf_I2_J,axiom,
! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
=> ( pp(aa_nat_bool(P,X2))
<=> pp(aa_nat_bool(P2,X2)) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
=> ( pp(aa_nat_bool(Q,X2))
<=> pp(aa_nat_bool(Q3,X2)) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( ( pp(aa_nat_bool(P,X3))
| pp(aa_nat_bool(Q,X3)) )
<=> ( pp(aa_nat_bool(P2,X3))
| pp(aa_nat_bool(Q3,X3)) ) ) ) ) ) ).
% minf(2)
tff(fact_856_minf_I2_J,axiom,
! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
=> ( pp(aa_int_bool(Q,X2))
<=> pp(aa_int_bool(Q3,X2)) ) )
=> ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( ( pp(aa_int_bool(P,X3))
| pp(aa_int_bool(Q,X3)) )
<=> ( pp(aa_int_bool(P2,X3))
| pp(aa_int_bool(Q3,X3)) ) ) ) ) ) ).
% minf(2)
tff(fact_857_minf_I1_J,axiom,
! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q3: fun_Co8822979462182099027r_bool] :
( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z3))
=> ( pp(aa_Code_integer_bool(P,X2))
<=> pp(aa_Code_integer_bool(P2,X2)) ) )
=> ( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z3))
=> ( pp(aa_Code_integer_bool(Q,X2))
<=> pp(aa_Code_integer_bool(Q3,X2)) ) )
=> ? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> ( ( pp(aa_Code_integer_bool(P,X3))
& pp(aa_Code_integer_bool(Q,X3)) )
<=> ( pp(aa_Code_integer_bool(P2,X3))
& pp(aa_Code_integer_bool(Q3,X3)) ) ) ) ) ) ).
% minf(1)
tff(fact_858_minf_I1_J,axiom,
! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
=> ( pp(aa_rat_bool(P,X2))
<=> pp(aa_rat_bool(P2,X2)) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z3))
=> ( pp(aa_rat_bool(Q,X2))
<=> pp(aa_rat_bool(Q3,X2)) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( ( pp(aa_rat_bool(P,X3))
& pp(aa_rat_bool(Q,X3)) )
<=> ( pp(aa_rat_bool(P2,X3))
& pp(aa_rat_bool(Q3,X3)) ) ) ) ) ) ).
% minf(1)
tff(fact_859_minf_I1_J,axiom,
! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
=> ( pp(aa_num_bool(P,X2))
<=> pp(aa_num_bool(P2,X2)) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z3))
=> ( pp(aa_num_bool(Q,X2))
<=> pp(aa_num_bool(Q3,X2)) ) )
=> ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( ( pp(aa_num_bool(P,X3))
& pp(aa_num_bool(Q,X3)) )
<=> ( pp(aa_num_bool(P2,X3))
& pp(aa_num_bool(Q3,X3)) ) ) ) ) ) ).
% minf(1)
tff(fact_860_minf_I1_J,axiom,
! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
=> ( pp(aa_nat_bool(P,X2))
<=> pp(aa_nat_bool(P2,X2)) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z3))
=> ( pp(aa_nat_bool(Q,X2))
<=> pp(aa_nat_bool(Q3,X2)) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( ( pp(aa_nat_bool(P,X3))
& pp(aa_nat_bool(Q,X3)) )
<=> ( pp(aa_nat_bool(P2,X3))
& pp(aa_nat_bool(Q3,X3)) ) ) ) ) ) ).
% minf(1)
tff(fact_861_minf_I1_J,axiom,
! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
=> ( pp(aa_int_bool(Q,X2))
<=> pp(aa_int_bool(Q3,X2)) ) )
=> ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( ( pp(aa_int_bool(P,X3))
& pp(aa_int_bool(Q,X3)) )
<=> ( pp(aa_int_bool(P2,X3))
& pp(aa_int_bool(Q3,X3)) ) ) ) ) ) ).
% minf(1)
tff(fact_862_pinf_I7_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,T),X3)) ) ).
% pinf(7)
tff(fact_863_pinf_I7_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X3)) ) ).
% pinf(7)
tff(fact_864_pinf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X3)) ) ).
% pinf(7)
tff(fact_865_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X3)) ) ).
% pinf(7)
tff(fact_866_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3)) ) ).
% pinf(7)
tff(fact_867_pinf_I5_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),T)) ) ).
% pinf(5)
tff(fact_868_pinf_I5_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),T)) ) ).
% pinf(5)
tff(fact_869_pinf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),T)) ) ).
% pinf(5)
tff(fact_870_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),T)) ) ).
% pinf(5)
tff(fact_871_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T)) ) ).
% pinf(5)
tff(fact_872_pinf_I4_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(4)
tff(fact_873_pinf_I4_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(4)
tff(fact_874_pinf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(4)
tff(fact_875_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(4)
tff(fact_876_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(4)
tff(fact_877_pinf_I3_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(3)
tff(fact_878_pinf_I3_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(3)
tff(fact_879_pinf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(3)
tff(fact_880_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(3)
tff(fact_881_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( X3 != T ) ) ).
% pinf(3)
tff(fact_882_pinf_I2_J,axiom,
! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q3: fun_Co8822979462182099027r_bool] :
( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X2))
=> ( pp(aa_Code_integer_bool(P,X2))
<=> pp(aa_Code_integer_bool(P2,X2)) ) )
=> ( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X2))
=> ( pp(aa_Code_integer_bool(Q,X2))
<=> pp(aa_Code_integer_bool(Q3,X2)) ) )
=> ? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> ( ( pp(aa_Code_integer_bool(P,X3))
| pp(aa_Code_integer_bool(Q,X3)) )
<=> ( pp(aa_Code_integer_bool(P2,X3))
| pp(aa_Code_integer_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(2)
tff(fact_883_pinf_I2_J,axiom,
! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
=> ( pp(aa_rat_bool(P,X2))
<=> pp(aa_rat_bool(P2,X2)) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
=> ( pp(aa_rat_bool(Q,X2))
<=> pp(aa_rat_bool(Q3,X2)) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( ( pp(aa_rat_bool(P,X3))
| pp(aa_rat_bool(Q,X3)) )
<=> ( pp(aa_rat_bool(P2,X3))
| pp(aa_rat_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(2)
tff(fact_884_pinf_I2_J,axiom,
! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
=> ( pp(aa_num_bool(P,X2))
<=> pp(aa_num_bool(P2,X2)) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
=> ( pp(aa_num_bool(Q,X2))
<=> pp(aa_num_bool(Q3,X2)) ) )
=> ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( ( pp(aa_num_bool(P,X3))
| pp(aa_num_bool(Q,X3)) )
<=> ( pp(aa_num_bool(P2,X3))
| pp(aa_num_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(2)
tff(fact_885_pinf_I2_J,axiom,
! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
=> ( pp(aa_nat_bool(P,X2))
<=> pp(aa_nat_bool(P2,X2)) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
=> ( pp(aa_nat_bool(Q,X2))
<=> pp(aa_nat_bool(Q3,X2)) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( ( pp(aa_nat_bool(P,X3))
| pp(aa_nat_bool(Q,X3)) )
<=> ( pp(aa_nat_bool(P2,X3))
| pp(aa_nat_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(2)
tff(fact_886_pinf_I2_J,axiom,
! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
=> ( pp(aa_int_bool(Q,X2))
<=> pp(aa_int_bool(Q3,X2)) ) )
=> ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( ( pp(aa_int_bool(P,X3))
| pp(aa_int_bool(Q,X3)) )
<=> ( pp(aa_int_bool(P2,X3))
| pp(aa_int_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(2)
tff(fact_887_pinf_I1_J,axiom,
! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q3: fun_Co8822979462182099027r_bool] :
( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X2))
=> ( pp(aa_Code_integer_bool(P,X2))
<=> pp(aa_Code_integer_bool(P2,X2)) ) )
=> ( ? [Z3: code_integer] :
! [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X2))
=> ( pp(aa_Code_integer_bool(Q,X2))
<=> pp(aa_Code_integer_bool(Q3,X2)) ) )
=> ? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> ( ( pp(aa_Code_integer_bool(P,X3))
& pp(aa_Code_integer_bool(Q,X3)) )
<=> ( pp(aa_Code_integer_bool(P2,X3))
& pp(aa_Code_integer_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(1)
tff(fact_888_pinf_I1_J,axiom,
! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
=> ( pp(aa_rat_bool(P,X2))
<=> pp(aa_rat_bool(P2,X2)) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X2))
=> ( pp(aa_rat_bool(Q,X2))
<=> pp(aa_rat_bool(Q3,X2)) ) )
=> ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( ( pp(aa_rat_bool(P,X3))
& pp(aa_rat_bool(Q,X3)) )
<=> ( pp(aa_rat_bool(P2,X3))
& pp(aa_rat_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(1)
tff(fact_889_pinf_I1_J,axiom,
! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
=> ( pp(aa_num_bool(P,X2))
<=> pp(aa_num_bool(P2,X2)) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X2))
=> ( pp(aa_num_bool(Q,X2))
<=> pp(aa_num_bool(Q3,X2)) ) )
=> ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( ( pp(aa_num_bool(P,X3))
& pp(aa_num_bool(Q,X3)) )
<=> ( pp(aa_num_bool(P2,X3))
& pp(aa_num_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(1)
tff(fact_890_pinf_I1_J,axiom,
! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
=> ( pp(aa_nat_bool(P,X2))
<=> pp(aa_nat_bool(P2,X2)) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X2))
=> ( pp(aa_nat_bool(Q,X2))
<=> pp(aa_nat_bool(Q3,X2)) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( ( pp(aa_nat_bool(P,X3))
& pp(aa_nat_bool(Q,X3)) )
<=> ( pp(aa_nat_bool(P2,X3))
& pp(aa_nat_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(1)
tff(fact_891_pinf_I1_J,axiom,
! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
=> ( pp(aa_int_bool(Q,X2))
<=> pp(aa_int_bool(Q3,X2)) ) )
=> ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( ( pp(aa_int_bool(P,X3))
& pp(aa_int_bool(Q,X3)) )
<=> ( pp(aa_int_bool(P2,X3))
& pp(aa_int_bool(Q3,X3)) ) ) ) ) ) ).
% pinf(1)
tff(fact_892_lt__ex,axiom,
! [X: rat] :
? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X)) ).
% lt_ex
tff(fact_893_lt__ex,axiom,
! [X: int] :
? [Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y2),X)) ).
% lt_ex
tff(fact_894_gt__ex,axiom,
! [X: rat] :
? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_12)) ).
% gt_ex
tff(fact_895_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_12)) ).
% gt_ex
tff(fact_896_gt__ex,axiom,
! [X: int] :
? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_12)) ).
% gt_ex
tff(fact_897_dense,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ? [Z2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y)) ) ) ).
% dense
tff(fact_898_less__imp__neq,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_899_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_900_less__imp__neq,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_901_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_902_less__imp__neq,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_903_order_Oasym,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ).
% order.asym
tff(fact_904_order_Oasym,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).
% order.asym
tff(fact_905_order_Oasym,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).
% order.asym
tff(fact_906_order_Oasym,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).
% order.asym
tff(fact_907_order_Oasym,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).
% order.asym
tff(fact_908_ord__eq__less__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A = B )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),C)) ) ) ).
% ord_eq_less_trans
tff(fact_909_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).
% ord_eq_less_trans
tff(fact_910_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).
% ord_eq_less_trans
tff(fact_911_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).
% ord_eq_less_trans
tff(fact_912_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).
% ord_eq_less_trans
tff(fact_913_ord__less__eq__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( ( B = C )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),C)) ) ) ).
% ord_less_eq_trans
tff(fact_914_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ( B = C )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).
% ord_less_eq_trans
tff(fact_915_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( ( B = C )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).
% ord_less_eq_trans
tff(fact_916_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( ( B = C )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).
% ord_less_eq_trans
tff(fact_917_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( ( B = C )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).
% ord_less_eq_trans
tff(fact_918_less__induct,axiom,
! [P: fun_nat_bool,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
=> pp(aa_nat_bool(P,Y3)) )
=> pp(aa_nat_bool(P,X2)) )
=> pp(aa_nat_bool(P,A)) ) ).
% less_induct
tff(fact_919_antisym__conv3,axiom,
! [Y: code_integer,X: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X))
=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_920_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_921_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_922_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_923_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_924_linorder__cases,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( ( X != Y )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).
% linorder_cases
tff(fact_925_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( ( X != Y )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).
% linorder_cases
tff(fact_926_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( ( X != Y )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).
% linorder_cases
tff(fact_927_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( ( X != Y )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).
% linorder_cases
tff(fact_928_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( ( X != Y )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).
% linorder_cases
tff(fact_929_dual__order_Oasym,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% dual_order.asym
tff(fact_930_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% dual_order.asym
tff(fact_931_dual__order_Oasym,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).
% dual_order.asym
tff(fact_932_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% dual_order.asym
tff(fact_933_dual__order_Oasym,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% dual_order.asym
tff(fact_934_dual__order_Oirrefl,axiom,
! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),A)) ).
% dual_order.irrefl
tff(fact_935_dual__order_Oirrefl,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).
% dual_order.irrefl
tff(fact_936_dual__order_Oirrefl,axiom,
! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).
% dual_order.irrefl
tff(fact_937_dual__order_Oirrefl,axiom,
! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).
% dual_order.irrefl
tff(fact_938_dual__order_Oirrefl,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).
% dual_order.irrefl
tff(fact_939_exists__least__iff,axiom,
! [P: fun_nat_bool] :
( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
<=> ? [N3: nat] :
( pp(aa_nat_bool(P,N3))
& ! [M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N3))
=> ~ pp(aa_nat_bool(P,M5)) ) ) ) ).
% exists_least_iff
tff(fact_940_linorder__less__wlog,axiom,
! [P: fun_Co8310303586393106575r_bool,A: code_integer,B: code_integer] :
( ! [A2: code_integer,B2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A2),B2)) )
=> ( ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A2),A2))
=> ( ! [A2: code_integer,B2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,B2),A2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A2),B2)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A),B)) ) ) ) ).
% linorder_less_wlog
tff(fact_941_linorder__less__wlog,axiom,
! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
( ! [A2: rat,B2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B2)) )
=> ( ! [A2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),A2))
=> ( ! [A2: rat,B2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B2),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A2),B2)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ) ).
% linorder_less_wlog
tff(fact_942_linorder__less__wlog,axiom,
! [P: fun_num_fun_num_bool,A: num,B: num] :
( ! [A2: num,B2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
=> pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) )
=> ( ! [A2: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A2),A2))
=> ( ! [A2: num,B2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(P,B2),A2))
=> pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) )
=> pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ) ).
% linorder_less_wlog
tff(fact_943_linorder__less__wlog,axiom,
! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) )
=> ( ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),A2))
=> ( ! [A2: nat,B2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B2),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).
% linorder_less_wlog
tff(fact_944_linorder__less__wlog,axiom,
! [P: fun_int_fun_int_bool,A: int,B: int] :
( ! [A2: int,B2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) )
=> ( ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A2),A2))
=> ( ! [A2: int,B2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(P,B2),A2))
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ) ).
% linorder_less_wlog
tff(fact_945_order_Ostrict__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),C)) ) ) ).
% order.strict_trans
tff(fact_946_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).
% order.strict_trans
tff(fact_947_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).
% order.strict_trans
tff(fact_948_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).
% order.strict_trans
tff(fact_949_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).
% order.strict_trans
tff(fact_950_not__less__iff__gr__or__eq,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X))
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_951_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_952_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_953_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_954_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_955_dual__order_Ostrict__trans,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A)) ) ) ).
% dual_order.strict_trans
tff(fact_956_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).
% dual_order.strict_trans
tff(fact_957_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).
% dual_order.strict_trans
tff(fact_958_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).
% dual_order.strict_trans
tff(fact_959_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).
% dual_order.strict_trans
tff(fact_960_order_Ostrict__implies__not__eq,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_961_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_962_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_963_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_964_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_965_dual__order_Ostrict__implies__not__eq,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_966_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_967_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_968_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_969_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_970_linorder__neqE,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).
% linorder_neqE
tff(fact_971_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).
% linorder_neqE
tff(fact_972_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).
% linorder_neqE
tff(fact_973_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).
% linorder_neqE
tff(fact_974_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).
% linorder_neqE
tff(fact_975_order__less__asym,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).
% order_less_asym
tff(fact_976_order__less__asym,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).
% order_less_asym
tff(fact_977_order__less__asym,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).
% order_less_asym
tff(fact_978_order__less__asym,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).
% order_less_asym
tff(fact_979_order__less__asym,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).
% order_less_asym
tff(fact_980_linorder__neq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).
% linorder_neq_iff
tff(fact_981_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).
% linorder_neq_iff
tff(fact_982_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).
% linorder_neq_iff
tff(fact_983_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).
% linorder_neq_iff
tff(fact_984_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).
% linorder_neq_iff
tff(fact_985_order__less__asym_H,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ).
% order_less_asym'
tff(fact_986_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).
% order_less_asym'
tff(fact_987_order__less__asym_H,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).
% order_less_asym'
tff(fact_988_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).
% order_less_asym'
tff(fact_989_order__less__asym_H,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).
% order_less_asym'
tff(fact_990_order__less__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z)) ) ) ).
% order_less_trans
tff(fact_991_order__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).
% order_less_trans
tff(fact_992_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).
% order_less_trans
tff(fact_993_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).
% order_less_trans
tff(fact_994_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).
% order_less_trans
tff(fact_995_ord__eq__less__subst,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(F,B) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_996_ord__eq__less__subst,axiom,
! [A: rat,F: fun_Code_integer_rat,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_rat(F,B) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_Code_integer_rat(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_997_ord__eq__less__subst,axiom,
! [A: num,F: fun_Code_integer_num,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_num(F,B) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F,X2)),aa_Code_integer_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_Code_integer_num(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_998_ord__eq__less__subst,axiom,
! [A: nat,F: fun_Code_integer_nat,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_nat(F,B) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_Code_integer_nat(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_999_ord__eq__less__subst,axiom,
! [A: int,F: fun_Code_integer_int,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_int(F,B) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_Code_integer_int(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_1000_ord__eq__less__subst,axiom,
! [A: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
( ( A = aa_rat_Code_integer(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_1001_ord__eq__less__subst,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( ( A = aa_rat_rat(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_1002_ord__eq__less__subst,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( ( A = aa_rat_num(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_1003_ord__eq__less__subst,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( ( A = aa_rat_nat(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_1004_ord__eq__less__subst,axiom,
! [A: int,F: fun_rat_int,B: rat,C: rat] :
( ( A = aa_rat_int(F,B) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).
% ord_eq_less_subst
tff(fact_1005_ord__less__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( ( aa_Cod1783974652533396728nteger(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1006_ord__less__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_rat,C: rat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( ( aa_Code_integer_rat(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1007_ord__less__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_num,C: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( ( aa_Code_integer_num(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F,X2)),aa_Code_integer_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1008_ord__less__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_nat,C: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( ( aa_Code_integer_nat(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1009_ord__less__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_int,C: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( ( aa_Code_integer_int(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1010_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ( aa_rat_Code_integer(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1011_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ( aa_rat_rat(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1012_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ( aa_rat_num(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1013_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ( aa_rat_nat(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1014_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ( aa_rat_int(F,B) = C )
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).
% ord_less_eq_subst
tff(fact_1015_order__less__irrefl,axiom,
! [X: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),X)) ).
% order_less_irrefl
tff(fact_1016_order__less__irrefl,axiom,
! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).
% order_less_irrefl
tff(fact_1017_order__less__irrefl,axiom,
! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).
% order_less_irrefl
tff(fact_1018_order__less__irrefl,axiom,
! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).
% order_less_irrefl
tff(fact_1019_order__less__irrefl,axiom,
! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).
% order_less_irrefl
tff(fact_1020_order__less__subst1,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1021_order__less__subst1,axiom,
! [A: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1022_order__less__subst1,axiom,
! [A: code_integer,F: fun_num_Code_integer,B: num,C: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F,X2)),aa_num_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1023_order__less__subst1,axiom,
! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_nat_Code_integer(F,B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_nat_Code_integer(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1024_order__less__subst1,axiom,
! [A: code_integer,F: fun_int_Code_integer,B: int,C: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_int_Code_integer(F,B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_int_Code_integer(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1025_order__less__subst1,axiom,
! [A: rat,F: fun_Code_integer_rat,B: code_integer,C: code_integer] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_Code_integer_rat(F,B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_Code_integer_rat(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1026_order__less__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1027_order__less__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1028_order__less__subst1,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1029_order__less__subst1,axiom,
! [A: rat,F: fun_int_rat,B: int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).
% order_less_subst1
tff(fact_1030_order__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1031_order__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_rat,C: rat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1032_order__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_num,C: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F,X2)),aa_Code_integer_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1033_order__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_nat,C: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1034_order__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_int,C: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1035_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1036_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1037_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1038_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1039_order__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).
% order_less_subst2
tff(fact_1040_order__less__not__sym,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).
% order_less_not_sym
tff(fact_1041_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).
% order_less_not_sym
tff(fact_1042_order__less__not__sym,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).
% order_less_not_sym
tff(fact_1043_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).
% order_less_not_sym
tff(fact_1044_order__less__not__sym,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).
% order_less_not_sym
tff(fact_1045_order__less__imp__triv,axiom,
! [X: code_integer,Y: code_integer,P: bool] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X))
=> pp(P) ) ) ).
% order_less_imp_triv
tff(fact_1046_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P: bool] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
=> pp(P) ) ) ).
% order_less_imp_triv
tff(fact_1047_order__less__imp__triv,axiom,
! [X: num,Y: num,P: bool] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
=> pp(P) ) ) ).
% order_less_imp_triv
tff(fact_1048_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
=> pp(P) ) ) ).
% order_less_imp_triv
tff(fact_1049_order__less__imp__triv,axiom,
! [X: int,Y: int,P: bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
=> pp(P) ) ) ).
% order_less_imp_triv
tff(fact_1050_linorder__less__linear,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
| ( X = Y )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).
% linorder_less_linear
tff(fact_1051_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
| ( X = Y )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).
% linorder_less_linear
tff(fact_1052_linorder__less__linear,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
| ( X = Y )
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).
% linorder_less_linear
tff(fact_1053_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
| ( X = Y )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).
% linorder_less_linear
tff(fact_1054_linorder__less__linear,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
| ( X = Y )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).
% linorder_less_linear
tff(fact_1055_order__less__imp__not__eq,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1056_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1057_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1058_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1059_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1060_order__less__imp__not__eq2,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1061_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1062_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1063_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1064_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1065_order__less__imp__not__less,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).
% order_less_imp_not_less
tff(fact_1066_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).
% order_less_imp_not_less
tff(fact_1067_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).
% order_less_imp_not_less
tff(fact_1068_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).
% order_less_imp_not_less
tff(fact_1069_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).
% order_less_imp_not_less
tff(fact_1070_verit__comp__simplify1_I1_J,axiom,
! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1071_verit__comp__simplify1_I1_J,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1072_verit__comp__simplify1_I1_J,axiom,
! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1073_verit__comp__simplify1_I1_J,axiom,
! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1074_verit__comp__simplify1_I1_J,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1075_floor__less__cancel,axiom,
! [X: rat,Y: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).
% floor_less_cancel
tff(fact_1076_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).
% zless_nat_eq_int_zless
tff(fact_1077_nat__mono__iff,axiom,
! [Z: int,W: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).
% nat_mono_iff
tff(fact_1078_power__int__strict__increasing,axiom,
! [N: int,N4: int,A: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(A),N4))) ) ) ).
% power_int_strict_increasing
tff(fact_1079_zero__less__power__int,axiom,
! [X: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(power_int_rat(X),N))) ) ).
% zero_less_power_int
tff(fact_1080_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% less_imp_of_nat_less
tff(fact_1081_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).
% less_imp_of_nat_less
tff(fact_1082_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ) ).
% less_imp_of_nat_less
tff(fact_1083_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).
% less_imp_of_nat_less
tff(fact_1084_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N))) ) ).
% less_imp_of_nat_less
tff(fact_1085_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_1086_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_1087_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_1088_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_1089_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_1090_nat__int__comparison_I2_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).
% nat_int_comparison(2)
tff(fact_1091_power__int__strict__decreasing,axiom,
! [N: int,N4: int,A: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(A),N4)),aa_int_rat(power_int_rat(A),N))) ) ) ) ).
% power_int_strict_decreasing
tff(fact_1092_power__int__strict__antimono,axiom,
! [A: rat,B: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),zero_zero_int))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(B),N)),aa_int_rat(power_int_rat(A),N))) ) ) ) ).
% power_int_strict_antimono
tff(fact_1093_ceiling__less__cancel,axiom,
! [X: rat,Y: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).
% ceiling_less_cancel
tff(fact_1094_one__less__power__int,axiom,
! [A: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(power_int_rat(A),N))) ) ) ).
% one_less_power_int
tff(fact_1095_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).
% nat_less_eq_zless
tff(fact_1096_nat__less__as__int,axiom,
! [X3: nat,Xa: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Xa))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).
% nat_less_as_int
tff(fact_1097_power__int__antimono,axiom,
! [A: rat,B: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),zero_zero_int))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(B),N)),aa_int_rat(power_int_rat(A),N))) ) ) ) ).
% power_int_antimono
tff(fact_1098_power__int__strict__mono,axiom,
! [A: rat,B: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(B),N))) ) ) ) ).
% power_int_strict_mono
tff(fact_1099_power__int__le__imp__less__exp,axiom,
! [X: rat,M: int,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ) ) ).
% power_int_le_imp_less_exp
tff(fact_1100_nat__less__iff,axiom,
! [W: int,M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).
% nat_less_iff
tff(fact_1101_less__ceiling__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).
% less_ceiling_iff
tff(fact_1102_verit__comp__simplify1_I3_J,axiom,
! [B4: code_integer,A4: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B4),A4))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A4),B4)) ) ).
% verit_comp_simplify1(3)
tff(fact_1103_verit__comp__simplify1_I3_J,axiom,
! [B4: rat,A4: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B4),A4))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A4),B4)) ) ).
% verit_comp_simplify1(3)
tff(fact_1104_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A4: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B4),A4))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A4),B4)) ) ).
% verit_comp_simplify1(3)
tff(fact_1105_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B4),A4))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B4)) ) ).
% verit_comp_simplify1(3)
tff(fact_1106_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A4: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),A4))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B4)) ) ).
% verit_comp_simplify1(3)
tff(fact_1107_leD,axiom,
! [Y: code_integer,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).
% leD
tff(fact_1108_leD,axiom,
! [Y: set_nat,X: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
=> ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y)) ) ).
% leD
tff(fact_1109_leD,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).
% leD
tff(fact_1110_leD,axiom,
! [Y: num,X: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).
% leD
tff(fact_1111_leD,axiom,
! [Y: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).
% leD
tff(fact_1112_leD,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).
% leD
tff(fact_1113_leI,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X)) ) ).
% leI
tff(fact_1114_leI,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).
% leI
tff(fact_1115_leI,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).
% leI
tff(fact_1116_leI,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).
% leI
tff(fact_1117_leI,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).
% leI
tff(fact_1118_nless__le,axiom,
! [A: code_integer,B: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
<=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
| ( A = B ) ) ) ).
% nless_le
tff(fact_1119_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ~ pp(aa_set_nat_bool(ord_less_set_nat(A),B))
<=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
| ( A = B ) ) ) ).
% nless_le
tff(fact_1120_nless__le,axiom,
! [A: rat,B: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
| ( A = B ) ) ) ).
% nless_le
tff(fact_1121_nless__le,axiom,
! [A: num,B: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
<=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
| ( A = B ) ) ) ).
% nless_le
tff(fact_1122_nless__le,axiom,
! [A: nat,B: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
| ( A = B ) ) ) ).
% nless_le
tff(fact_1123_nless__le,axiom,
! [A: int,B: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
| ( A = B ) ) ) ).
% nless_le
tff(fact_1124_antisym__conv1,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_1125_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_1126_antisym__conv1,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_1127_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_1128_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_1129_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_1130_antisym__conv2,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_1131_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_1132_antisym__conv2,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_1133_antisym__conv2,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_1134_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_1135_antisym__conv2,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_1136_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).
% dense_ge
tff(fact_1137_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Z)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).
% dense_le
tff(fact_1138_less__le__not__le,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
& ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X)) ) ) ).
% less_le_not_le
tff(fact_1139_less__le__not__le,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
& ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).
% less_le_not_le
tff(fact_1140_less__le__not__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
& ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).
% less_le_not_le
tff(fact_1141_less__le__not__le,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
& ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).
% less_le_not_le
tff(fact_1142_less__le__not__le,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
& ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).
% less_le_not_le
tff(fact_1143_less__le__not__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
& ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).
% less_le_not_le
tff(fact_1144_not__le__imp__less,axiom,
! [Y: code_integer,X: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).
% not_le_imp_less
tff(fact_1145_not__le__imp__less,axiom,
! [Y: rat,X: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).
% not_le_imp_less
tff(fact_1146_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).
% not_le_imp_less
tff(fact_1147_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).
% not_le_imp_less
tff(fact_1148_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).
% not_le_imp_less
tff(fact_1149_order_Oorder__iff__strict,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_1150_order_Oorder__iff__strict,axiom,
! [A: set_nat,B: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
<=> ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_1151_order_Oorder__iff__strict,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_1152_order_Oorder__iff__strict,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_1153_order_Oorder__iff__strict,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_1154_order_Oorder__iff__strict,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_1155_order_Ostrict__iff__order,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_1156_order_Ostrict__iff__order,axiom,
! [A: set_nat,B: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_1157_order_Ostrict__iff__order,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_1158_order_Ostrict__iff__order,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_1159_order_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_1160_order_Ostrict__iff__order,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_1161_order_Ostrict__trans1,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),C)) ) ) ).
% order.strict_trans1
tff(fact_1162_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> ( pp(aa_set_nat_bool(ord_less_set_nat(B),C))
=> pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).
% order.strict_trans1
tff(fact_1163_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).
% order.strict_trans1
tff(fact_1164_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).
% order.strict_trans1
tff(fact_1165_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).
% order.strict_trans1
tff(fact_1166_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).
% order.strict_trans1
tff(fact_1167_order_Ostrict__trans2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),C)) ) ) ).
% order.strict_trans2
tff(fact_1168_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
=> pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).
% order.strict_trans2
tff(fact_1169_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).
% order.strict_trans2
tff(fact_1170_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).
% order.strict_trans2
tff(fact_1171_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).
% order.strict_trans2
tff(fact_1172_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).
% order.strict_trans2
tff(fact_1173_order_Ostrict__iff__not,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
& ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ).
% order.strict_iff_not
tff(fact_1174_order_Ostrict__iff__not,axiom,
! [A: set_nat,B: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
& ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).
% order.strict_iff_not
tff(fact_1175_order_Ostrict__iff__not,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
& ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% order.strict_iff_not
tff(fact_1176_order_Ostrict__iff__not,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
& ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).
% order.strict_iff_not
tff(fact_1177_order_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
& ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).
% order.strict_iff_not
tff(fact_1178_order_Ostrict__iff__not,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
& ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).
% order.strict_iff_not
tff(fact_1179_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
=> ( ! [W2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W2)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).
% dense_ge_bounded
tff(fact_1180_dense__le__bounded,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( ! [W2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).
% dense_le_bounded
tff(fact_1181_dual__order_Oorder__iff__strict,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1182_dual__order_Oorder__iff__strict,axiom,
! [B: set_nat,A: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
<=> ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1183_dual__order_Oorder__iff__strict,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1184_dual__order_Oorder__iff__strict,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1185_dual__order_Oorder__iff__strict,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1186_dual__order_Oorder__iff__strict,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1187_dual__order_Ostrict__iff__order,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1188_dual__order_Ostrict__iff__order,axiom,
! [B: set_nat,A: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1189_dual__order_Ostrict__iff__order,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1190_dual__order_Ostrict__iff__order,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1191_dual__order_Ostrict__iff__order,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1192_dual__order_Ostrict__iff__order,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1193_dual__order_Ostrict__trans1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A)) ) ) ).
% dual_order.strict_trans1
tff(fact_1194_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
=> ( pp(aa_set_nat_bool(ord_less_set_nat(C),B))
=> pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).
% dual_order.strict_trans1
tff(fact_1195_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).
% dual_order.strict_trans1
tff(fact_1196_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).
% dual_order.strict_trans1
tff(fact_1197_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).
% dual_order.strict_trans1
tff(fact_1198_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).
% dual_order.strict_trans1
tff(fact_1199_dual__order_Ostrict__trans2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A)) ) ) ).
% dual_order.strict_trans2
tff(fact_1200_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
=> pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).
% dual_order.strict_trans2
tff(fact_1201_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).
% dual_order.strict_trans2
tff(fact_1202_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).
% dual_order.strict_trans2
tff(fact_1203_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).
% dual_order.strict_trans2
tff(fact_1204_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).
% dual_order.strict_trans2
tff(fact_1205_dual__order_Ostrict__iff__not,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
& ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).
% dual_order.strict_iff_not
tff(fact_1206_dual__order_Ostrict__iff__not,axiom,
! [B: set_nat,A: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
& ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).
% dual_order.strict_iff_not
tff(fact_1207_dual__order_Ostrict__iff__not,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
& ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% dual_order.strict_iff_not
tff(fact_1208_dual__order_Ostrict__iff__not,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
& ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).
% dual_order.strict_iff_not
tff(fact_1209_dual__order_Ostrict__iff__not,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
& ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% dual_order.strict_iff_not
tff(fact_1210_dual__order_Ostrict__iff__not,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% dual_order.strict_iff_not
tff(fact_1211_order_Ostrict__implies__order,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% order.strict_implies_order
tff(fact_1212_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).
% order.strict_implies_order
tff(fact_1213_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% order.strict_implies_order
tff(fact_1214_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).
% order.strict_implies_order
tff(fact_1215_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% order.strict_implies_order
tff(fact_1216_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% order.strict_implies_order
tff(fact_1217_dual__order_Ostrict__implies__order,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).
% dual_order.strict_implies_order
tff(fact_1218_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ).
% dual_order.strict_implies_order
tff(fact_1219_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% dual_order.strict_implies_order
tff(fact_1220_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).
% dual_order.strict_implies_order
tff(fact_1221_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).
% dual_order.strict_implies_order
tff(fact_1222_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% dual_order.strict_implies_order
tff(fact_1223_order__le__less,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_1224_order__le__less,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
<=> ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_1225_order__le__less,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_1226_order__le__less,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_1227_order__le__less,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_1228_order__le__less,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_1229_order__less__le,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_1230_order__less__le,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_1231_order__less__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_1232_order__less__le,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_1233_order__less__le,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_1234_order__less__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_1235_linorder__not__le,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).
% linorder_not_le
tff(fact_1236_linorder__not__le,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).
% linorder_not_le
tff(fact_1237_linorder__not__le,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).
% linorder_not_le
tff(fact_1238_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).
% linorder_not_le
tff(fact_1239_linorder__not__le,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).
% linorder_not_le
tff(fact_1240_linorder__not__less,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X)) ) ).
% linorder_not_less
tff(fact_1241_linorder__not__less,axiom,
! [X: rat,Y: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).
% linorder_not_less
tff(fact_1242_linorder__not__less,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).
% linorder_not_less
tff(fact_1243_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).
% linorder_not_less
tff(fact_1244_linorder__not__less,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).
% linorder_not_less
tff(fact_1245_order__less__imp__le,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ).
% order_less_imp_le
tff(fact_1246_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).
% order_less_imp_le
tff(fact_1247_order__less__imp__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% order_less_imp_le
tff(fact_1248_order__less__imp__le,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).
% order_less_imp_le
tff(fact_1249_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).
% order_less_imp_le
tff(fact_1250_order__less__imp__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).
% order_less_imp_le
tff(fact_1251_order__le__neq__trans,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( ( A != B )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).
% order_le_neq_trans
tff(fact_1252_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> ( ( A != B )
=> pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).
% order_le_neq_trans
tff(fact_1253_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ( A != B )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% order_le_neq_trans
tff(fact_1254_order__le__neq__trans,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( ( A != B )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).
% order_le_neq_trans
tff(fact_1255_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( ( A != B )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% order_le_neq_trans
tff(fact_1256_order__le__neq__trans,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( ( A != B )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% order_le_neq_trans
tff(fact_1257_order__neq__le__trans,axiom,
! [A: code_integer,B: code_integer] :
( ( A != B )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).
% order_neq_le_trans
tff(fact_1258_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
=> pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).
% order_neq_le_trans
tff(fact_1259_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% order_neq_le_trans
tff(fact_1260_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).
% order_neq_le_trans
tff(fact_1261_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% order_neq_le_trans
tff(fact_1262_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% order_neq_le_trans
tff(fact_1263_order__le__less__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z)) ) ) ).
% order_le_less_trans
tff(fact_1264_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> ( pp(aa_set_nat_bool(ord_less_set_nat(Y),Z))
=> pp(aa_set_nat_bool(ord_less_set_nat(X),Z)) ) ) ).
% order_le_less_trans
tff(fact_1265_order__le__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).
% order_le_less_trans
tff(fact_1266_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).
% order_le_less_trans
tff(fact_1267_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).
% order_le_less_trans
tff(fact_1268_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).
% order_le_less_trans
tff(fact_1269_order__less__le__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),Z))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z)) ) ) ).
% order_less_le_trans
tff(fact_1270_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z))
=> pp(aa_set_nat_bool(ord_less_set_nat(X),Z)) ) ) ).
% order_less_le_trans
tff(fact_1271_order__less__le__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).
% order_less_le_trans
tff(fact_1272_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).
% order_less_le_trans
tff(fact_1273_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).
% order_less_le_trans
tff(fact_1274_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).
% order_less_le_trans
tff(fact_1275_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(F,B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1276_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_rat_Code_integer(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1277_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_num_Code_integer,B: num,C: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_num_Code_integer(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F,X2)),aa_num_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1278_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(F,B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_nat_Code_integer(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1279_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_int_Code_integer,B: int,C: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_int_Code_integer(F,B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_int_Code_integer(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1280_order__le__less__subst1,axiom,
! [A: rat,F: fun_Code_integer_rat,B: code_integer,C: code_integer] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_Code_integer_rat(F,B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_Code_integer_rat(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1281_order__le__less__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1282_order__le__less__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1283_order__le__less__subst1,axiom,
! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1284_order__le__less__subst1,axiom,
! [A: rat,F: fun_int_rat,B: int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).
% order_le_less_subst1
tff(fact_1285_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1286_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1287_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1288_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1289_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_int,C: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1290_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_Code_integer,C: code_integer] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(F,X2)),aa_num_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1291_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1292_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1293_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_nat,C: nat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1294_order__le__less__subst2,axiom,
! [A: num,B: num,F: fun_num_int,C: int] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).
% order_le_less_subst2
tff(fact_1295_order__less__le__subst1,axiom,
! [A: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_rat_Code_integer(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1296_order__less__le__subst1,axiom,
! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1297_order__less__le__subst1,axiom,
! [A: num,F: fun_rat_num,B: rat,C: rat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1298_order__less__le__subst1,axiom,
! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1299_order__less__le__subst1,axiom,
! [A: int,F: fun_rat_int,B: rat,C: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1300_order__less__le__subst1,axiom,
! [A: code_integer,F: fun_num_Code_integer,B: num,C: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(F,X2)),aa_num_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_num_Code_integer(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1301_order__less__le__subst1,axiom,
! [A: rat,F: fun_num_rat,B: num,C: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1302_order__less__le__subst1,axiom,
! [A: num,F: fun_num_num,B: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y2))) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1303_order__less__le__subst1,axiom,
! [A: nat,F: fun_num_nat,B: num,C: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1304_order__less__le__subst1,axiom,
! [A: int,F: fun_num_int,B: num,C: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).
% order_less_le_subst1
tff(fact_1305_order__less__le__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1306_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X2)),aa_rat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1307_order__less__le__subst2,axiom,
! [A: num,B: num,F: fun_num_Code_integer,C: code_integer] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F,X2)),aa_num_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1308_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,B)),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1309_order__less__le__subst2,axiom,
! [A: int,B: int,F: fun_int_Code_integer,C: code_integer] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F,B)),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(F,Y2))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1310_order__less__le__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_rat,C: rat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,B)),C))
=> ( ! [X2: code_integer,Y2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X2)),aa_Code_integer_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1311_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
=> ( ! [X2: rat,Y2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1312_order__less__le__subst2,axiom,
! [A: num,B: num,F: fun_num_rat,C: rat] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
=> ( ! [X2: num,Y2: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1313_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1314_order__less__le__subst2,axiom,
! [A: int,B: int,F: fun_int_rat,C: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C))
=> ( ! [X2: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).
% order_less_le_subst2
tff(fact_1315_linorder__le__less__linear,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).
% linorder_le_less_linear
tff(fact_1316_linorder__le__less__linear,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).
% linorder_le_less_linear
tff(fact_1317_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).
% linorder_le_less_linear
tff(fact_1318_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).
% linorder_le_less_linear
tff(fact_1319_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).
% linorder_le_less_linear
tff(fact_1320_order__le__imp__less__or__eq,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1321_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1322_order__le__imp__less__or__eq,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1323_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1324_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1325_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1326_minf_I8_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,T),X3)) ) ).
% minf(8)
tff(fact_1327_minf_I8_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X3)) ) ).
% minf(8)
tff(fact_1328_minf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X3)) ) ).
% minf(8)
tff(fact_1329_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X3)) ) ).
% minf(8)
tff(fact_1330_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3)) ) ).
% minf(8)
tff(fact_1331_minf_I6_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z2))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),T)) ) ).
% minf(6)
tff(fact_1332_minf_I6_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),T)) ) ).
% minf(6)
tff(fact_1333_minf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),T)) ) ).
% minf(6)
tff(fact_1334_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),T)) ) ).
% minf(6)
tff(fact_1335_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T)) ) ).
% minf(6)
tff(fact_1336_pinf_I8_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,T),X3)) ) ).
% pinf(8)
tff(fact_1337_pinf_I8_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X3)) ) ).
% pinf(8)
tff(fact_1338_pinf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X3)) ) ).
% pinf(8)
tff(fact_1339_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X3)) ) ).
% pinf(8)
tff(fact_1340_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3)) ) ).
% pinf(8)
tff(fact_1341_pinf_I6_J,axiom,
! [T: code_integer] :
? [Z2: code_integer] :
! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X3))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),T)) ) ).
% pinf(6)
tff(fact_1342_pinf_I6_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),T)) ) ).
% pinf(6)
tff(fact_1343_pinf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),T)) ) ).
% pinf(6)
tff(fact_1344_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),T)) ) ).
% pinf(6)
tff(fact_1345_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T)) ) ).
% pinf(6)
tff(fact_1346_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
tff(fact_1347_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
tff(fact_1348_not__less__zero,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% not_less_zero
tff(fact_1349_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% gr_zeroI
tff(fact_1350_less__numeral__extra_I3_J,axiom,
~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),zero_z2226904508553997617atural)) ).
% less_numeral_extra(3)
tff(fact_1351_less__numeral__extra_I3_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).
% less_numeral_extra(3)
tff(fact_1352_less__numeral__extra_I3_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).
% less_numeral_extra(3)
tff(fact_1353_less__numeral__extra_I3_J,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).
% less_numeral_extra(3)
tff(fact_1354_less__numeral__extra_I3_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).
% less_numeral_extra(3)
tff(fact_1355_add__mono__thms__linordered__field_I5_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_1356_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_1357_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_1358_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_1359_add__mono__thms__linordered__field_I2_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( ( I = J )
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_1360_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_1361_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_1362_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_1363_add__mono__thms__linordered__field_I1_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J))
& ( K = L ) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_1364_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
& ( K = L ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_1365_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& ( K = L ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_1366_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
& ( K = L ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_1367_add__strict__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D))) ) ) ).
% add_strict_mono
tff(fact_1368_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).
% add_strict_mono
tff(fact_1369_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_strict_mono
tff(fact_1370_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_strict_mono
tff(fact_1371_add__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))) ) ).
% add_strict_left_mono
tff(fact_1372_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).
% add_strict_left_mono
tff(fact_1373_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).
% add_strict_left_mono
tff(fact_1374_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% add_strict_left_mono
tff(fact_1375_add__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))) ) ).
% add_strict_right_mono
tff(fact_1376_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ).
% add_strict_right_mono
tff(fact_1377_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).
% add_strict_right_mono
tff(fact_1378_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).
% add_strict_right_mono
tff(fact_1379_add__less__imp__less__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% add_less_imp_less_left
tff(fact_1380_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_imp_less_left
tff(fact_1381_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_imp_less_left
tff(fact_1382_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_imp_less_left
tff(fact_1383_add__less__imp__less__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% add_less_imp_less_right
tff(fact_1384_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_imp_less_right
tff(fact_1385_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_imp_less_right
tff(fact_1386_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_imp_less_right
tff(fact_1387_less__numeral__extra_I4_J,axiom,
~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),one_one_Code_natural)) ).
% less_numeral_extra(4)
tff(fact_1388_less__numeral__extra_I4_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer)) ).
% less_numeral_extra(4)
tff(fact_1389_less__numeral__extra_I4_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).
% less_numeral_extra(4)
tff(fact_1390_less__numeral__extra_I4_J,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).
% less_numeral_extra(4)
tff(fact_1391_less__numeral__extra_I4_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).
% less_numeral_extra(4)
tff(fact_1392_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).
% bot_nat_0.extremum_strict
tff(fact_1393_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% gr0I
tff(fact_1394_not__gr0,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ( N = zero_zero_nat ) ) ).
% not_gr0
tff(fact_1395_not__less0,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% not_less0
tff(fact_1396_less__zeroE,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% less_zeroE
tff(fact_1397_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
tff(fact_1398_infinite__descent0,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( ~ pp(aa_nat_bool(P,N2))
=> ? [M6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N2))
& ~ pp(aa_nat_bool(P,M6)) ) ) )
=> pp(aa_nat_bool(P,N)) ) ) ).
% infinite_descent0
tff(fact_1399_reals__Archimedean2,axiom,
! [X: rat] :
? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).
% reals_Archimedean2
tff(fact_1400_less__int__code_I1_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).
% less_int_code(1)
tff(fact_1401_exists__leI,axiom,
! [N: nat,P: fun_nat_bool] :
( ( ! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),N))
=> ~ pp(aa_nat_bool(P,N5)) )
=> pp(aa_nat_bool(P,N)) )
=> ? [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N))
& pp(aa_nat_bool(P,N6)) ) ) ).
% exists_leI
tff(fact_1402_nat__less__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& ( M != N ) ) ) ).
% nat_less_le
tff(fact_1403_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% less_imp_le_nat
tff(fact_1404_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( M = N ) ) ) ).
% le_eq_less_or_eq
tff(fact_1405_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( M = N ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% less_or_eq_imp_le
tff(fact_1406_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( ( M != N )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% le_neq_implies_less
tff(fact_1407_less__mono__imp__le__mono,axiom,
! [F: fun_nat_nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).
% less_mono_imp_le_mono
tff(fact_1408_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).
% add_lessD1
tff(fact_1409_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).
% add_less_mono
tff(fact_1410_not__add__less1,axiom,
! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).
% not_add_less1
tff(fact_1411_not__add__less2,axiom,
! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).
% not_add_less2
tff(fact_1412_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).
% add_less_mono1
tff(fact_1413_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
% trans_less_add1
tff(fact_1414_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
% trans_less_add2
tff(fact_1415_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% less_add_eq_less
tff(fact_1416_ex__less__of__int,axiom,
! [X: rat] :
? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).
% ex_less_of_int
tff(fact_1417_ex__of__int__less,axiom,
! [X: rat] :
? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)) ).
% ex_of_int_less
tff(fact_1418_of__nat__floor,axiom,
! [R: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R)))),R)) ) ).
% of_nat_floor
tff(fact_1419_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).
% nat_le_eq_zle
tff(fact_1420_split__nat,axiom,
! [P: fun_nat_bool,I: int] :
( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
<=> ( ! [N3: nat] :
( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
=> pp(aa_nat_bool(P,N3)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
=> pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).
% split_nat
tff(fact_1421_of__int__pos,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))) ) ).
% of_int_pos
tff(fact_1422_of__int__pos,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% of_int_pos
tff(fact_1423_of__int__pos,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).
% of_int_pos
tff(fact_1424_zero__less__imp__eq__int,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
& ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).
% zero_less_imp_eq_int
tff(fact_1425_pos__int__cases,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ~ ! [N2: nat] :
( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).
% pos_int_cases
tff(fact_1426_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).
% of_nat_less_of_int_iff
tff(fact_1427_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).
% of_nat_less_of_int_iff
tff(fact_1428_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).
% of_nat_less_of_int_iff
tff(fact_1429_zero__le__power__int,axiom,
! [X: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(power_int_rat(X),N))) ) ).
% zero_le_power_int
tff(fact_1430_power__int__le__imp__le__exp,axiom,
! [X: rat,M: int,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N)) ) ) ) ).
% power_int_le_imp_le_exp
tff(fact_1431_floor__split,axiom,
! [P: fun_int_bool,T: rat] :
( pp(aa_int_bool(P,aa_rat_int(archim3151403230148437115or_rat,T)))
<=> ! [I3: int] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I3)),T))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat))) )
=> pp(aa_int_bool(P,I3)) ) ) ).
% floor_split
tff(fact_1432_floor__eq__iff,axiom,
! [X: rat,A: int] :
( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat))) ) ) ).
% floor_eq_iff
tff(fact_1433_floor__unique,axiom,
! [Z: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)))
=> ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).
% floor_unique
tff(fact_1434_less__floor__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).
% less_floor_iff
tff(fact_1435_power__int__not__zero,axiom,
! [X: rat,N: int] :
( ( ( X != zero_zero_rat )
| ( N = zero_zero_int ) )
=> ( aa_int_rat(power_int_rat(X),N) != zero_zero_rat ) ) ).
% power_int_not_zero
tff(fact_1436_floor__le__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).
% floor_le_iff
tff(fact_1437_floor__correct,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)))) ) ).
% floor_correct
tff(fact_1438_floor__mono,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))) ) ).
% floor_mono
tff(fact_1439_nat__zero__as__int,axiom,
zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).
% nat_zero_as_int
tff(fact_1440_of__int__floor__le,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X)) ).
% of_int_floor_le
tff(fact_1441_nat__mono,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).
% nat_mono
tff(fact_1442_ex__nat,axiom,
! [P: fun_nat_bool] :
( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
<=> ? [X6: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
& pp(aa_nat_bool(P,aa_int_nat(nat2,X6))) ) ) ).
% ex_nat
tff(fact_1443_all__nat,axiom,
! [P: fun_nat_bool] :
( ! [X_13: nat] : pp(aa_nat_bool(P,X_13))
<=> ! [X6: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
=> pp(aa_nat_bool(P,aa_int_nat(nat2,X6))) ) ) ).
% all_nat
tff(fact_1444_eq__nat__nat__iff,axiom,
! [Z: int,Z4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
=> ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
<=> ( Z = Z4 ) ) ) ) ).
% eq_nat_nat_iff
tff(fact_1445_nat__one__as__int,axiom,
one_one_nat = aa_int_nat(nat2,one_one_int) ).
% nat_one_as_int
tff(fact_1446_add__less__le__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D))) ) ) ).
% add_less_le_mono
tff(fact_1447_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).
% add_less_le_mono
tff(fact_1448_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_less_le_mono
tff(fact_1449_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_less_le_mono
tff(fact_1450_add__le__less__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D))) ) ) ).
% add_le_less_mono
tff(fact_1451_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).
% add_le_less_mono
tff(fact_1452_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_le_less_mono
tff(fact_1453_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_le_less_mono
tff(fact_1454_add__mono__thms__linordered__field_I3_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1455_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1456_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1457_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1458_add__mono__thms__linordered__field_I4_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),J))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L))) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1459_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1460_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1461_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1462_add__less__zeroD,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),zero_z3403309356797280102nteger)) ) ) ).
% add_less_zeroD
tff(fact_1463_add__less__zeroD,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).
% add_less_zeroD
tff(fact_1464_add__less__zeroD,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).
% add_less_zeroD
tff(fact_1465_add__neg__neg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% add_neg_neg
tff(fact_1466_add__neg__neg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% add_neg_neg
tff(fact_1467_add__neg__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).
% add_neg_neg
tff(fact_1468_add__neg__neg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_neg_neg
tff(fact_1469_add__neg__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_neg_neg
tff(fact_1470_add__pos__pos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))) ) ) ).
% add_pos_pos
tff(fact_1471_add__pos__pos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))) ) ) ).
% add_pos_pos
tff(fact_1472_add__pos__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).
% add_pos_pos
tff(fact_1473_add__pos__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_pos_pos
tff(fact_1474_add__pos__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_pos_pos
tff(fact_1475_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ~ ! [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_1476_pos__add__strict,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C))) ) ) ).
% pos_add_strict
tff(fact_1477_pos__add__strict,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C))) ) ) ).
% pos_add_strict
tff(fact_1478_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).
% pos_add_strict
tff(fact_1479_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% pos_add_strict
tff(fact_1480_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% pos_add_strict
tff(fact_1481_zero__less__one,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural)) ).
% zero_less_one
tff(fact_1482_zero__less__one,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).
% zero_less_one
tff(fact_1483_zero__less__one,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).
% zero_less_one
tff(fact_1484_zero__less__one,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
% zero_less_one
tff(fact_1485_zero__less__one,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).
% zero_less_one
tff(fact_1486_not__one__less__zero,axiom,
~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),zero_z2226904508553997617atural)) ).
% not_one_less_zero
tff(fact_1487_not__one__less__zero,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).
% not_one_less_zero
tff(fact_1488_not__one__less__zero,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).
% not_one_less_zero
tff(fact_1489_not__one__less__zero,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
% not_one_less_zero
tff(fact_1490_not__one__less__zero,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).
% not_one_less_zero
tff(fact_1491_less__numeral__extra_I1_J,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural)) ).
% less_numeral_extra(1)
tff(fact_1492_less__numeral__extra_I1_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).
% less_numeral_extra(1)
tff(fact_1493_less__numeral__extra_I1_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).
% less_numeral_extra(1)
tff(fact_1494_less__numeral__extra_I1_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
% less_numeral_extra(1)
tff(fact_1495_less__numeral__extra_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).
% less_numeral_extra(1)
tff(fact_1496_less__add__one,axiom,
! [A: code_natural] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural))) ).
% less_add_one
tff(fact_1497_less__add__one,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer))) ).
% less_add_one
tff(fact_1498_less__add__one,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat))) ).
% less_add_one
tff(fact_1499_less__add__one,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).
% less_add_one
tff(fact_1500_less__add__one,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).
% less_add_one
tff(fact_1501_add__mono1,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),one_one_Code_natural))) ) ).
% add_mono1
tff(fact_1502_add__mono1,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),one_one_Code_integer))) ) ).
% add_mono1
tff(fact_1503_add__mono1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat))) ) ).
% add_mono1
tff(fact_1504_add__mono1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat))) ) ).
% add_mono1
tff(fact_1505_add__mono1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int))) ) ).
% add_mono1
tff(fact_1506_floor__le__ceiling,axiom,
! [X: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X))) ).
% floor_le_ceiling
tff(fact_1507_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).
% of_nat_less_0_iff
tff(fact_1508_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).
% of_nat_less_0_iff
tff(fact_1509_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)) ).
% of_nat_less_0_iff
tff(fact_1510_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).
% of_nat_less_0_iff
tff(fact_1511_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),zero_z2226904508553997617atural)) ).
% of_nat_less_0_iff
tff(fact_1512_ex__least__nat__le,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,N))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ? [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
=> ~ pp(aa_nat_bool(P,I4)) )
& pp(aa_nat_bool(P,K3)) ) ) ) ).
% ex_least_nat_le
tff(fact_1513_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ? [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
& ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).
% less_imp_add_positive
tff(fact_1514_mono__nat__linear__lb,axiom,
! [F: fun_nat_nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).
% mono_nat_linear_lb
tff(fact_1515_zless__add1__eq,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
| ( W = Z ) ) ) ).
% zless_add1_eq
tff(fact_1516_int__gr__induct,axiom,
! [K: int,I: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_gr_induct
tff(fact_1517_power__int__0__left__If,axiom,
! [M: int] :
( ( ( M = zero_zero_int )
=> ( aa_int_rat(power_int_rat(zero_zero_rat),M) = one_one_rat ) )
& ( ( M != zero_zero_int )
=> ( aa_int_rat(power_int_rat(zero_zero_rat),M) = zero_zero_rat ) ) ) ).
% power_int_0_left_If
tff(fact_1518_power__int__increasing,axiom,
! [N: int,N4: int,A: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(A),N)),aa_int_rat(power_int_rat(A),N4))) ) ) ).
% power_int_increasing
tff(fact_1519_le__floor__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).
% le_floor_iff
tff(fact_1520_le__floor__add,axiom,
! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)))) ).
% le_floor_add
tff(fact_1521_int__add__floor,axiom,
! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ).
% int_add_floor
tff(fact_1522_floor__add__int,axiom,
! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ).
% floor_add_int
tff(fact_1523_nat__le__iff,axiom,
! [X: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% nat_le_iff
tff(fact_1524_nat__0__le,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).
% nat_0_le
tff(fact_1525_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
<=> ( ( M = aa_int_nat(nat2,Z) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).
% int_eq_iff
tff(fact_1526_of__nat__ceiling,axiom,
! [R: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R))))) ).
% of_nat_ceiling
tff(fact_1527_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_1528_add__neg__nonpos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% add_neg_nonpos
tff(fact_1529_add__neg__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% add_neg_nonpos
tff(fact_1530_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).
% add_neg_nonpos
tff(fact_1531_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_neg_nonpos
tff(fact_1532_add__neg__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_neg_nonpos
tff(fact_1533_add__nonneg__pos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_1534_add__nonneg__pos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_1535_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_1536_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_1537_add__nonneg__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_1538_add__nonpos__neg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% add_nonpos_neg
tff(fact_1539_add__nonpos__neg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% add_nonpos_neg
tff(fact_1540_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).
% add_nonpos_neg
tff(fact_1541_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_nonpos_neg
tff(fact_1542_add__nonpos__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_nonpos_neg
tff(fact_1543_add__pos__nonneg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_1544_add__pos__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_1545_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_1546_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_1547_add__pos__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_1548_add__strict__increasing,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C))) ) ) ).
% add_strict_increasing
tff(fact_1549_add__strict__increasing,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C))) ) ) ).
% add_strict_increasing
tff(fact_1550_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).
% add_strict_increasing
tff(fact_1551_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_strict_increasing
tff(fact_1552_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_strict_increasing
tff(fact_1553_add__strict__increasing2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_1554_add__strict__increasing2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_1555_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_1556_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_1557_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_1558_zero__less__two,axiom,
pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),one_one_Code_natural))) ).
% zero_less_two
tff(fact_1559_zero__less__two,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer))) ).
% zero_less_two
tff(fact_1560_zero__less__two,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).
% zero_less_two
tff(fact_1561_zero__less__two,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% zero_less_two
tff(fact_1562_zero__less__two,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).
% zero_less_two
tff(fact_1563_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% int_one_le_iff_zero_less
tff(fact_1564_odd__less__0__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% odd_less_0_iff
tff(fact_1565_add1__zle__eq,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% add1_zle_eq
tff(fact_1566_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)) ) ).
% zless_imp_add1_zle
tff(fact_1567_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_1568_power__int__mono,axiom,
! [X: rat,Y: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(power_int_rat(X),N)),aa_int_rat(power_int_rat(Y),N))) ) ) ) ).
% power_int_mono
tff(fact_1569_one__le__power__int,axiom,
! [X: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(power_int_rat(X),N))) ) ) ).
% one_le_power_int
tff(fact_1570_one__add__floor,axiom,
! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) ).
% one_add_floor
tff(fact_1571_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M = aa_int_nat(nat2,W) )
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
tff(fact_1572_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( aa_int_nat(nat2,W) = M )
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
tff(fact_1573_le__nat__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).
% le_nat_iff
tff(fact_1574_nat__add__distrib,axiom,
! [Z: int,Z4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).
% nat_add_distrib
tff(fact_1575_ceiling__altdef,axiom,
! [X: rat] :
( ( ( X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
=> ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) )
& ( ( X != aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
=> ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ) ) ) ).
% ceiling_altdef
tff(fact_1576_floor__exists,axiom,
! [X: rat] :
? [Z2: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).
% floor_exists
tff(fact_1577_floor__exists1,axiom,
! [X: rat] :
? [X2: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int))))
& ! [Y3: int] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
=> ( Y3 = X2 ) ) ) ).
% floor_exists1
tff(fact_1578_discrete,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),B)) ) ).
% discrete
tff(fact_1579_discrete,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).
% discrete
tff(fact_1580_discrete,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).
% discrete
tff(fact_1581_field__le__epsilon,axiom,
! [X: rat,Y: rat] :
( ! [E3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% field_le_epsilon
tff(fact_1582_floor__add,axiom,
! [X: rat,Y: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
=> ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
=> ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int) ) ) ) ).
% floor_add
tff(fact_1583_complete__interval,axiom,
! [A: nat,B: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(P,A))
=> ( ~ pp(aa_nat_bool(P,B))
=> ? [C3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C3))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C3),B))
& ! [X3: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),C3)) )
=> pp(aa_nat_bool(P,X3)) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),D2)) )
=> pp(aa_nat_bool(P,X2)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D2),C3)) ) ) ) ) ) ).
% complete_interval
tff(fact_1584_complete__interval,axiom,
! [A: int,B: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(P,A))
=> ( ~ pp(aa_int_bool(P,B))
=> ? [C3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C3),B))
& ! [X3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),C3)) )
=> pp(aa_int_bool(P,X3)) )
& ! [D2: int] :
( ! [X2: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),D2)) )
=> pp(aa_int_bool(P,X2)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D2),C3)) ) ) ) ) ) ).
% complete_interval
tff(fact_1585_zero__le__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)) ) ).
% zero_le_ceiling
tff(fact_1586_ceiling__less__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ) ).
% ceiling_less_zero
tff(fact_1587_time__array__of__list,axiom,
! [Xs: list_nat,H: heap_e7401611519738050253t_unit] : time_time_array_nat(array_of_list_nat(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat) ).
% time_array_of_list
tff(fact_1588_time__array__of__list,axiom,
! [Xs: list_int,H: heap_e7401611519738050253t_unit] : time_time_array_int(array_of_list_int(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat) ).
% time_array_of_list
tff(fact_1589_time__array__of__list,axiom,
! [Xs: list_bool,H: heap_e7401611519738050253t_unit] : time_time_array_bool(array_of_list_bool(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat) ).
% time_array_of_list
tff(fact_1590_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).
% Suc_nat_eq_nat_zadd1
tff(fact_1591_gen__length__def,axiom,
! [N: nat,Xs: list_nat] : aa_list_nat_nat(gen_length_nat(N),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)) ).
% gen_length_def
tff(fact_1592_gen__length__def,axiom,
! [N: nat,Xs: list_int] : aa_list_int_nat(gen_length_int(N),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_int_nat(size_size_list_int,Xs)) ).
% gen_length_def
tff(fact_1593_gen__length__def,axiom,
! [N: nat,Xs: list_bool] : aa_list_bool_nat(gen_length_bool(N),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)) ).
% gen_length_def
tff(fact_1594_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_1595_verit__minus__simplify_I4_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).
% verit_minus_simplify(4)
tff(fact_1596_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_1597_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_1598_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_1599_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_1600_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_1601_add_Oinverse__inverse,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).
% add.inverse_inverse
tff(fact_1602_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_1603_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_1604_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
<=> ( X22 = Y22 ) ) ).
% nat.inject
tff(fact_1605_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% neg_le_iff_le
tff(fact_1606_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% neg_le_iff_le
tff(fact_1607_neg__le__iff__le,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% neg_le_iff_le
tff(fact_1608_add_Oinverse__neutral,axiom,
aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
% add.inverse_neutral
tff(fact_1609_add_Oinverse__neutral,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% add.inverse_neutral
tff(fact_1610_add_Oinverse__neutral,axiom,
aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).
% add.inverse_neutral
tff(fact_1611_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_1612_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_1613_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_1614_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_1615_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_1616_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_1617_equal__neg__zero,axiom,
! [A: int] :
( ( A = aa_int_int(uminus_uminus_int,A) )
<=> ( A = zero_zero_int ) ) ).
% equal_neg_zero
tff(fact_1618_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
tff(fact_1619_equal__neg__zero,axiom,
! [A: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,A) )
<=> ( A = zero_zero_rat ) ) ).
% equal_neg_zero
tff(fact_1620_neg__equal__zero,axiom,
! [A: int] :
( ( aa_int_int(uminus_uminus_int,A) = A )
<=> ( A = zero_zero_int ) ) ).
% neg_equal_zero
tff(fact_1621_neg__equal__zero,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
tff(fact_1622_neg__equal__zero,axiom,
! [A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = A )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_zero
tff(fact_1623_neg__less__iff__less,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% neg_less_iff_less
tff(fact_1624_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% neg_less_iff_less
tff(fact_1625_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% neg_less_iff_less
tff(fact_1626_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_1627_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% minus_add_distrib
tff(fact_1628_minus__add__distrib,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).
% minus_add_distrib
tff(fact_1629_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_1630_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = B ).
% minus_add_cancel
tff(fact_1631_minus__add__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).
% minus_add_cancel
tff(fact_1632_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_1633_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) = B ).
% add_minus_cancel
tff(fact_1634_add__minus__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).
% add_minus_cancel
tff(fact_1635_lessI,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).
% lessI
tff(fact_1636_Suc__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).
% Suc_mono
tff(fact_1637_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_less_eq
tff(fact_1638_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).
% Suc_le_mono
tff(fact_1639_add__Suc__right,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
% add_Suc_right
tff(fact_1640_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
<=> ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
tff(fact_1641_of__int__minus,axiom,
! [Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z)) ).
% of_int_minus
tff(fact_1642_of__int__minus,axiom,
! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).
% of_int_minus
tff(fact_1643_of__int__minus,axiom,
! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).
% of_int_minus
tff(fact_1644_negative__zle,axiom,
! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).
% negative_zle
tff(fact_1645_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_less_eq_nonneg
tff(fact_1646_neg__less__eq__nonneg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% neg_less_eq_nonneg
tff(fact_1647_neg__less__eq__nonneg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% neg_less_eq_nonneg
tff(fact_1648_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% less_eq_neg_nonpos
tff(fact_1649_less__eq__neg__nonpos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% less_eq_neg_nonpos
tff(fact_1650_less__eq__neg__nonpos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% less_eq_neg_nonpos
tff(fact_1651_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_le_0_iff_le
tff(fact_1652_neg__le__0__iff__le,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% neg_le_0_iff_le
tff(fact_1653_neg__le__0__iff__le,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% neg_le_0_iff_le
tff(fact_1654_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% neg_0_le_iff_le
tff(fact_1655_neg__0__le__iff__le,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% neg_0_le_iff_le
tff(fact_1656_neg__0__le__iff__le,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% neg_0_le_iff_le
tff(fact_1657_neg__less__0__iff__less,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).
% neg_less_0_iff_less
tff(fact_1658_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_less_0_iff_less
tff(fact_1659_neg__less__0__iff__less,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% neg_less_0_iff_less
tff(fact_1660_neg__0__less__iff__less,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% neg_0_less_iff_less
tff(fact_1661_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% neg_0_less_iff_less
tff(fact_1662_neg__0__less__iff__less,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% neg_0_less_iff_less
tff(fact_1663_neg__less__pos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).
% neg_less_pos
tff(fact_1664_neg__less__pos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_less_pos
tff(fact_1665_neg__less__pos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% neg_less_pos
tff(fact_1666_less__neg__neg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% less_neg_neg
tff(fact_1667_less__neg__neg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% less_neg_neg
tff(fact_1668_less__neg__neg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% less_neg_neg
tff(fact_1669_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_1670_ab__left__minus,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).
% ab_left_minus
tff(fact_1671_ab__left__minus,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).
% ab_left_minus
tff(fact_1672_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_1673_add_Oright__inverse,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).
% add.right_inverse
tff(fact_1674_add_Oright__inverse,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).
% add.right_inverse
tff(fact_1675_zero__less__Suc,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).
% zero_less_Suc
tff(fact_1676_less__Suc0,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
<=> ( N = zero_zero_nat ) ) ).
% less_Suc0
tff(fact_1677_negative__zless,axiom,
! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).
% negative_zless
tff(fact_1678_nat__zminus__int,axiom,
! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).
% nat_zminus_int
tff(fact_1679_floor__uminus__of__int,axiom,
! [Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(uminus_uminus_int,Z) ).
% floor_uminus_of_int
tff(fact_1680_frac__of__int,axiom,
! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).
% frac_of_int
tff(fact_1681_dbl__inc__simps_I4_J,axiom,
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_1682_dbl__inc__simps_I4_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_inc_simps(4)
tff(fact_1683_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_1684_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_1685_add__neg__numeral__special_I8_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% add_neg_numeral_special(8)
tff(fact_1686_add__neg__numeral__special_I8_J,axiom,
aa_rat_rat(aa_rat_fun_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_1687_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_1688_add__neg__numeral__special_I7_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% add_neg_numeral_special(7)
tff(fact_1689_add__neg__numeral__special_I7_J,axiom,
aa_rat_rat(aa_rat_fun_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_1690_of__nat__Suc,axiom,
! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).
% of_nat_Suc
tff(fact_1691_of__nat__Suc,axiom,
! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).
% of_nat_Suc
tff(fact_1692_of__nat__Suc,axiom,
! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).
% of_nat_Suc
tff(fact_1693_of__nat__Suc,axiom,
! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).
% of_nat_Suc
tff(fact_1694_of__nat__Suc,axiom,
! [M: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(suc,M)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_nat_Code_natural(semiri3763490453095760265atural,M)) ).
% of_nat_Suc
tff(fact_1695_nat__1,axiom,
aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).
% nat_1
tff(fact_1696_one__less__nat__eq,axiom,
! [Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% one_less_nat_eq
tff(fact_1697_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
=> ( ( K != aa_nat_nat(suc,I) )
=> ~ ! [J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
=> ( K != aa_nat_nat(suc,J2) ) ) ) ) ).
% Nat.lessE
tff(fact_1698_Suc__lessD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_lessD
tff(fact_1699_Suc__lessE,axiom,
! [I: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
=> ~ ! [J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
=> ( K != aa_nat_nat(suc,J2) ) ) ) ).
% Suc_lessE
tff(fact_1700_Suc__lessI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( ( aa_nat_nat(suc,M) != N )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).
% Suc_lessI
tff(fact_1701_less__SucE,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( M = N ) ) ) ).
% less_SucE
tff(fact_1702_less__SucI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).
% less_SucI
tff(fact_1703_Ex__less__Suc,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
& pp(aa_nat_bool(P,I3)) )
<=> ( pp(aa_nat_bool(P,N))
| ? [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
& pp(aa_nat_bool(P,I3)) ) ) ) ).
% Ex_less_Suc
tff(fact_1704_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( M = N ) ) ) ).
% less_Suc_eq
tff(fact_1705_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% nat_neq_iff
tff(fact_1706_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).
% not_less_eq
tff(fact_1707_All__less__Suc,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
=> pp(aa_nat_bool(P,I3)) )
<=> ( pp(aa_nat_bool(P,N))
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
=> pp(aa_nat_bool(P,I3)) ) ) ) ).
% All_less_Suc
tff(fact_1708_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
<=> ? [M7: nat] :
( ( M = aa_nat_nat(suc,M7) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M7)) ) ) ).
% Suc_less_eq2
tff(fact_1709_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
=> ( M = N ) ) ) ).
% less_antisym
tff(fact_1710_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_less_SucD
tff(fact_1711_less__not__refl,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).
% less_not_refl
tff(fact_1712_less__not__refl2,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( M != N ) ) ).
% less_not_refl2
tff(fact_1713_less__not__refl3,axiom,
! [S: nat,T: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
% less_not_refl3
tff(fact_1714_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).
% less_trans_Suc
tff(fact_1715_less__Suc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
=> ( ! [I2: nat,J2: nat,K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K3)) ) ) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).
% less_Suc_induct
tff(fact_1716_less__irrefl__nat,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).
% less_irrefl_nat
tff(fact_1717_nat__less__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( ! [N2: nat] :
( ! [M6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N2))
=> pp(aa_nat_bool(P,M6)) )
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,N)) ) ).
% nat_less_induct
tff(fact_1718_infinite__descent,axiom,
! [P: fun_nat_bool,N: nat] :
( ! [N2: nat] :
( ~ pp(aa_nat_bool(P,N2))
=> ? [M6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N2))
& ~ pp(aa_nat_bool(P,M6)) ) )
=> pp(aa_nat_bool(P,N)) ) ).
% infinite_descent
tff(fact_1719_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).
% linorder_neqE_nat
tff(fact_1720_strict__inc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( ! [I2: nat] :
( ( J = aa_nat_nat(suc,I2) )
=> pp(aa_nat_bool(P,I2)) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
=> ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
=> pp(aa_nat_bool(P,I2)) ) )
=> pp(aa_nat_bool(P,I)) ) ) ) ).
% strict_inc_induct
tff(fact_1721_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
<=> ( N = M ) ) ) ).
% not_less_less_Suc_eq
tff(fact_1722_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_1723_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_1724_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_1725_n__not__Suc__n,axiom,
! [N: nat] : N != aa_nat_nat(suc,N) ).
% n_not_Suc_n
tff(fact_1726_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
% Suc_inject
tff(fact_1727_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_1728_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_1729_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_1730_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_1731_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_1732_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_1733_int__of__nat__induct,axiom,
! [P: fun_int_bool,Z: int] :
( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
=> ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
=> pp(aa_int_bool(P,Z)) ) ) ).
% int_of_nat_induct
tff(fact_1734_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
=> ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).
% int_cases
tff(fact_1735_ceiling__def,axiom,
! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X))) ).
% ceiling_def
tff(fact_1736_floor__minus,axiom,
! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim2889992004027027881ng_rat,X)) ).
% floor_minus
tff(fact_1737_ceiling__minus,axiom,
! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,X)) ).
% ceiling_minus
tff(fact_1738_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% le_imp_neg_le
tff(fact_1739_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).
% le_imp_neg_le
tff(fact_1740_le__imp__neg__le,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).
% le_imp_neg_le
tff(fact_1741_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).
% minus_le_iff
tff(fact_1742_minus__le__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).
% minus_le_iff
tff(fact_1743_minus__le__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).
% minus_le_iff
tff(fact_1744_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% le_minus_iff
tff(fact_1745_le__minus__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).
% le_minus_iff
tff(fact_1746_le__minus__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).
% le_minus_iff
tff(fact_1747_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).
% verit_negate_coefficient(2)
tff(fact_1748_verit__negate__coefficient_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% verit_negate_coefficient(2)
tff(fact_1749_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).
% verit_negate_coefficient(2)
tff(fact_1750_minus__less__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).
% minus_less_iff
tff(fact_1751_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).
% minus_less_iff
tff(fact_1752_minus__less__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).
% minus_less_iff
tff(fact_1753_less__minus__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).
% less_minus_iff
tff(fact_1754_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% less_minus_iff
tff(fact_1755_less__minus__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).
% less_minus_iff
tff(fact_1756_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_1757_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).
% add.inverse_distrib_swap
tff(fact_1758_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_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_1759_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_1760_group__cancel_Oneg1,axiom,
! [A3: code_integer,K: code_integer,A: code_integer] :
( ( A3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).
% group_cancel.neg1
tff(fact_1761_group__cancel_Oneg1,axiom,
! [A3: rat,K: rat,A: rat] :
( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
=> ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).
% group_cancel.neg1
tff(fact_1762_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_1763_is__num__normalize_I8_J,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).
% is_num_normalize(8)
tff(fact_1764_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_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_1765_one__neq__neg__one,axiom,
one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% one_neq_neg_one
tff(fact_1766_one__neq__neg__one,axiom,
one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% one_neq_neg_one
tff(fact_1767_one__neq__neg__one,axiom,
one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% one_neq_neg_one
tff(fact_1768_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_Code_integer,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N2)),aa_nat_Code_integer(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1769_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_rat,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1770_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_num,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1771_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_nat,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1772_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_int,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1773_lift__Suc__mono__less,axiom,
! [F: fun_nat_Code_integer,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N2)),aa_nat_Code_integer(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,N7))) ) ) ).
% lift_Suc_mono_less
tff(fact_1774_lift__Suc__mono__less,axiom,
! [F: fun_nat_rat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N7))) ) ) ).
% lift_Suc_mono_less
tff(fact_1775_lift__Suc__mono__less,axiom,
! [F: fun_nat_num,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N7))) ) ) ).
% lift_Suc_mono_less
tff(fact_1776_lift__Suc__mono__less,axiom,
! [F: fun_nat_nat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N7))) ) ) ).
% lift_Suc_mono_less
tff(fact_1777_lift__Suc__mono__less,axiom,
! [F: fun_nat_int,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N7))) ) ) ).
% lift_Suc_mono_less
tff(fact_1778_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).
% not0_implies_Suc
tff(fact_1779_Zero__not__Suc,axiom,
! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).
% Zero_not_Suc
tff(fact_1780_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).
% Zero_neq_Suc
tff(fact_1781_Suc__neq__Zero,axiom,
! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).
% Suc_neq_Zero
tff(fact_1782_zero__induct,axiom,
! [P: fun_nat_bool,K: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).
% zero_induct
tff(fact_1783_diff__induct,axiom,
! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),zero_zero_nat))
=> ( ! [Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y2)))
=> ( ! [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).
% diff_induct
tff(fact_1784_nat__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
=> pp(aa_nat_bool(P,N)) ) ) ).
% nat_induct
tff(fact_1785_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).
% old.nat.exhaust
tff(fact_1786_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat = aa_nat_nat(suc,X22) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
tff(fact_1787_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).
% old.nat.distinct(1)
tff(fact_1788_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).
% old.nat.distinct(2)
tff(fact_1789_nat_Odistinct_I1_J,axiom,
! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).
% nat.distinct(1)
tff(fact_1790_exists__least__lemma,axiom,
! [P: fun_nat_bool] :
( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
=> ? [N2: nat] :
( ~ pp(aa_nat_bool(P,N2))
& pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).
% exists_least_lemma
tff(fact_1791_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).
% less_Suc_eq_0_disj
tff(fact_1792_gr0__implies__Suc,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).
% gr0_implies_Suc
tff(fact_1793_All__less__Suc2,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
=> pp(aa_nat_bool(P,I3)) )
<=> ( pp(aa_nat_bool(P,zero_zero_nat))
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).
% All_less_Suc2
tff(fact_1794_gr0__conv__Suc,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).
% gr0_conv_Suc
tff(fact_1795_Ex__less__Suc2,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
& pp(aa_nat_bool(P,I3)) )
<=> ( pp(aa_nat_bool(P,zero_zero_nat))
| ? [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
& pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).
% Ex_less_Suc2
tff(fact_1796_uminus__int__code_I1_J,axiom,
aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
% uminus_int_code(1)
tff(fact_1797_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: fun_nat_fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),X2))
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y2),Z2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Z2)) ) )
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,N2),aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,M),N)) ) ) ) ) ).
% transitive_stepwise_le
tff(fact_1798_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(P,M))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_induct_at_least
tff(fact_1799_full__nat__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( ! [N2: nat] :
( ! [M6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M6)),N2))
=> pp(aa_nat_bool(P,M6)) )
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,N)) ) ).
% full_nat_induct
tff(fact_1800_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).
% not_less_eq_eq
tff(fact_1801_Suc__n__not__le__n,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).
% Suc_n_not_le_n
tff(fact_1802_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
| ( M = aa_nat_nat(suc,N) ) ) ) ).
% le_Suc_eq
tff(fact_1803_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M4))
=> ? [M3: nat] : M4 = aa_nat_nat(suc,M3) ) ).
% Suc_le_D
tff(fact_1804_le__SucI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).
% le_SucI
tff(fact_1805_le__SucE,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( M = aa_nat_nat(suc,N) ) ) ) ).
% le_SucE
tff(fact_1806_Suc__leD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% Suc_leD
tff(fact_1807_Suc__leI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).
% Suc_leI
tff(fact_1808_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_le_eq
tff(fact_1809_dec__induct,axiom,
! [I: nat,J: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(P,I))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
=> pp(aa_nat_bool(P,J)) ) ) ) ).
% dec_induct
tff(fact_1810_inc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(P,J))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
=> ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(P,N2)) ) ) )
=> pp(aa_nat_bool(P,I)) ) ) ) ).
% inc_induct
tff(fact_1811_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_le_lessD
tff(fact_1812_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
<=> ( N = M ) ) ) ).
% le_less_Suc_eq
tff(fact_1813_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% less_Suc_eq_le
tff(fact_1814_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).
% less_eq_Suc_le
tff(fact_1815_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).
% le_imp_less_Suc
tff(fact_1816_nat__in__between__eq_I1_J,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(suc,A))) )
<=> ( B = aa_nat_nat(suc,A) ) ) ).
% nat_in_between_eq(1)
tff(fact_1817_nat__in__between__eq_I2_J,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(suc,A))) )
<=> ( B = A ) ) ).
% nat_in_between_eq(2)
tff(fact_1818_add__Suc__shift,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).
% add_Suc_shift
tff(fact_1819_add__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
% add_Suc
tff(fact_1820_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_1821_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).
% less_imp_Suc_add
tff(fact_1822_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
<=> ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).
% less_iff_Suc_add
tff(fact_1823_less__add__Suc2,axiom,
! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).
% less_add_Suc2
tff(fact_1824_less__add__Suc1,axiom,
! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).
% less_add_Suc1
tff(fact_1825_less__natE,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ~ ! [Q4: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ) ).
% less_natE
tff(fact_1826_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
=> ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).
% int_cases2
tff(fact_1827_not__zle__0__negative,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).
% not_zle_0_negative
tff(fact_1828_negD,axiom,
! [X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
=> ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).
% negD
tff(fact_1829_negative__zless__0,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).
% negative_zless_0
tff(fact_1830_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_1831_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_1832_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = B )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_1833_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_1834_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_1835_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_1836_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_1837_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).
% add.inverse_unique
tff(fact_1838_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).
% add.inverse_unique
tff(fact_1839_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_1840_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).
% ab_group_add_class.ab_left_minus
tff(fact_1841_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_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_1842_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_1843_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
<=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% add_eq_0_iff
tff(fact_1844_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
<=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% add_eq_0_iff
tff(fact_1845_le__minus__one__simps_I4_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% le_minus_one_simps(4)
tff(fact_1846_le__minus__one__simps_I4_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% le_minus_one_simps(4)
tff(fact_1847_le__minus__one__simps_I4_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% le_minus_one_simps(4)
tff(fact_1848_le__minus__one__simps_I2_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).
% le_minus_one_simps(2)
tff(fact_1849_le__minus__one__simps_I2_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).
% le_minus_one_simps(2)
tff(fact_1850_le__minus__one__simps_I2_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).
% le_minus_one_simps(2)
tff(fact_1851_zero__neq__neg__one,axiom,
zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% zero_neq_neg_one
tff(fact_1852_zero__neq__neg__one,axiom,
zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% zero_neq_neg_one
tff(fact_1853_zero__neq__neg__one,axiom,
zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% zero_neq_neg_one
tff(fact_1854_less__minus__one__simps_I2_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).
% less_minus_one_simps(2)
tff(fact_1855_less__minus__one__simps_I2_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).
% less_minus_one_simps(2)
tff(fact_1856_less__minus__one__simps_I2_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).
% less_minus_one_simps(2)
tff(fact_1857_less__minus__one__simps_I4_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% less_minus_one_simps(4)
tff(fact_1858_less__minus__one__simps_I4_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% less_minus_one_simps(4)
tff(fact_1859_less__minus__one__simps_I4_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% less_minus_one_simps(4)
tff(fact_1860_lift__Suc__antimono__le,axiom,
! [F: fun_nat_set_nat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N7)),aa_nat_set_nat(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1861_lift__Suc__antimono__le,axiom,
! [F: fun_nat_rat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N7)),aa_nat_rat(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1862_lift__Suc__antimono__le,axiom,
! [F: fun_nat_num,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N7)),aa_nat_num(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1863_lift__Suc__antimono__le,axiom,
! [F: fun_nat_nat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N7)),aa_nat_nat(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1864_lift__Suc__antimono__le,axiom,
! [F: fun_nat_int,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N7)),aa_nat_int(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1865_lift__Suc__mono__le,axiom,
! [F: fun_nat_set_nat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N7))) ) ) ).
% lift_Suc_mono_le
tff(fact_1866_lift__Suc__mono__le,axiom,
! [F: fun_nat_rat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N7))) ) ) ).
% lift_Suc_mono_le
tff(fact_1867_lift__Suc__mono__le,axiom,
! [F: fun_nat_num,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N7))) ) ) ).
% lift_Suc_mono_le
tff(fact_1868_lift__Suc__mono__le,axiom,
! [F: fun_nat_nat,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N7))) ) ) ).
% lift_Suc_mono_le
tff(fact_1869_lift__Suc__mono__le,axiom,
! [F: fun_nat_int,N: nat,N7: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N7))) ) ) ).
% lift_Suc_mono_le
tff(fact_1870_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).
% of_nat_neq_0
tff(fact_1871_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).
% of_nat_neq_0
tff(fact_1872_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N)) != zero_z3403309356797280102nteger ).
% of_nat_neq_0
tff(fact_1873_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).
% of_nat_neq_0
tff(fact_1874_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(suc,N)) != zero_z2226904508553997617atural ).
% of_nat_neq_0
tff(fact_1875_nat__compl__induct_H,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N2))
=> pp(aa_nat_bool(P,Nn)) )
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
=> pp(aa_nat_bool(P,N)) ) ) ).
% nat_compl_induct'
tff(fact_1876_nat__compl__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N2))
=> pp(aa_nat_bool(P,Nn)) )
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
=> pp(aa_nat_bool(P,N)) ) ) ).
% nat_compl_induct
tff(fact_1877_ex__least__nat__less,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,N))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ? [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K3))
=> ~ pp(aa_nat_bool(P,I4)) )
& pp(aa_nat_bool(P,aa_nat_nat(suc,K3))) ) ) ) ).
% ex_least_nat_less
tff(fact_1878_one__is__add,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
<=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
% one_is_add
tff(fact_1879_add__is__1,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
% add_is_1
tff(fact_1880_One__nat__def,axiom,
one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).
% One_nat_def
tff(fact_1881_nat__induct__non__zero,axiom,
! [N: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(P,one_one_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_induct_non_zero
tff(fact_1882_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
=> ~ ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).
% int_cases4
tff(fact_1883_Suc__eq__plus1__left,axiom,
! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).
% Suc_eq_plus1_left
tff(fact_1884_plus__1__eq__Suc,axiom,
aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).
% plus_1_eq_Suc
tff(fact_1885_Suc__eq__plus1,axiom,
! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).
% Suc_eq_plus1
tff(fact_1886_not__int__zless__negative,axiom,
! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).
% not_int_zless_negative
tff(fact_1887_of__int__of__nat,axiom,
! [K: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_int(ring_1_of_int_int,K) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K)) ) ) ) ).
% of_int_of_nat
tff(fact_1888_of__int__of__nat,axiom,
! [K: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K)) ) ) ) ).
% of_int_of_nat
tff(fact_1889_of__int__of__nat,axiom,
! [K: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_rat(ring_1_of_int_rat,K) = aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K)) ) ) ) ).
% of_int_of_nat
tff(fact_1890_frac__ge__0,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).
% frac_ge_0
tff(fact_1891_frac__lt__1,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).
% frac_lt_1
tff(fact_1892_frac__1__eq,axiom,
! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).
% frac_1_eq
tff(fact_1893_le__minus__one__simps_I1_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).
% le_minus_one_simps(1)
tff(fact_1894_le__minus__one__simps_I1_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).
% le_minus_one_simps(1)
tff(fact_1895_le__minus__one__simps_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).
% le_minus_one_simps(1)
tff(fact_1896_le__minus__one__simps_I3_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% le_minus_one_simps(3)
tff(fact_1897_le__minus__one__simps_I3_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% le_minus_one_simps(3)
tff(fact_1898_le__minus__one__simps_I3_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% le_minus_one_simps(3)
tff(fact_1899_less__minus__one__simps_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).
% less_minus_one_simps(1)
tff(fact_1900_less__minus__one__simps_I1_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).
% less_minus_one_simps(1)
tff(fact_1901_less__minus__one__simps_I1_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).
% less_minus_one_simps(1)
tff(fact_1902_less__minus__one__simps_I3_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% less_minus_one_simps(3)
tff(fact_1903_less__minus__one__simps_I3_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% less_minus_one_simps(3)
tff(fact_1904_less__minus__one__simps_I3_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% less_minus_one_simps(3)
tff(fact_1905_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
=> ~ ! [N2: nat] :
( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).
% int_cases3
tff(fact_1906_int__zle__neg,axiom,
! [N: nat,M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
<=> ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
tff(fact_1907_nonpos__int__cases,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
=> ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).
% nonpos_int_cases
tff(fact_1908_negative__zle__0,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).
% negative_zle_0
tff(fact_1909_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(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_1910_int__Suc,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).
% int_Suc
tff(fact_1911_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_1912_neg__int__cases,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ~ ! [N2: nat] :
( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).
% neg_int_cases
tff(fact_1913_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_1914_frac__eq,axiom,
! [X: rat] :
( ( aa_rat_rat(archimedean_frac_rat,X) = X )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).
% frac_eq
tff(fact_1915_length__code,axiom,
size_size_list_nat = gen_length_nat(zero_zero_nat) ).
% length_code
tff(fact_1916_length__code,axiom,
size_size_list_int = gen_length_int(zero_zero_nat) ).
% length_code
tff(fact_1917_length__code,axiom,
size_size_list_bool = gen_length_bool(zero_zero_nat) ).
% length_code
tff(fact_1918_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ).
% compl_le_compl_iff
tff(fact_1919_power__int__minus__one__minus,axiom,
! [N: int] : aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_int_int(uminus_uminus_int,N)) = aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) ).
% power_int_minus_one_minus
tff(fact_1920_dbl__dec__simps_I2_J,axiom,
neg_nu3811975205180677377ec_int(zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).
% dbl_dec_simps(2)
tff(fact_1921_dbl__dec__simps_I2_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_dec_simps(2)
tff(fact_1922_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_1923_nat__descend__induct,axiom,
! [N: nat,P: fun_nat_bool,M: nat] :
( ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
=> pp(aa_nat_bool(P,K3)) )
=> ( ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
=> ( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),I4))
=> pp(aa_nat_bool(P,I4)) )
=> pp(aa_nat_bool(P,K3)) ) )
=> pp(aa_nat_bool(P,M)) ) ) ).
% nat_descend_induct
tff(fact_1924_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).
% list_decode.cases
tff(fact_1925_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y)) ) ).
% compl_le_swap2
tff(fact_1926_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X)))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).
% compl_le_swap1
tff(fact_1927_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).
% compl_mono
tff(fact_1928_dbl__dec__simps_I3_J,axiom,
neg_nu3811975205180677377ec_int(one_one_int) = one_one_int ).
% dbl_dec_simps(3)
tff(fact_1929_dbl__dec__simps_I3_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,one_one_Code_integer) = one_one_Code_integer ).
% dbl_dec_simps(3)
tff(fact_1930_dbl__dec__simps_I3_J,axiom,
aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).
% dbl_dec_simps(3)
tff(fact_1931_triangle__Suc,axiom,
! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).
% triangle_Suc
tff(fact_1932_bot__nat__0_Oordering__top__axioms,axiom,
pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_bg,aTP_Lamm_bh),zero_zero_nat)) ).
% bot_nat_0.ordering_top_axioms
tff(fact_1933_power__int__add__1_H,axiom,
! [X: rat,M: int] :
( ( ( X != zero_zero_rat )
| ( M != aa_int_int(uminus_uminus_int,one_one_int) ) )
=> ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),one_one_int)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_int_rat(power_int_rat(X),M)) ) ) ).
% power_int_add_1'
tff(fact_1934_power__int__add__1,axiom,
! [X: rat,M: int] :
( ( ( X != zero_zero_rat )
| ( M != aa_int_int(uminus_uminus_int,one_one_int) ) )
=> ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),one_one_int)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),M)),X) ) ) ).
% power_int_add_1
tff(fact_1935_length__ge__1__conv,axiom,
! [L: list_char] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_char(L)))
<=> ( L != nil_char ) ) ).
% length_ge_1_conv
tff(fact_1936_length__ge__1__conv,axiom,
! [L: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_nat_nat(size_size_list_nat,L)))
<=> ( L != nil_nat ) ) ).
% length_ge_1_conv
tff(fact_1937_length__ge__1__conv,axiom,
! [L: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_int_nat(size_size_list_int,L)))
<=> ( L != nil_int ) ) ).
% length_ge_1_conv
tff(fact_1938_length__ge__1__conv,axiom,
! [L: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_bool_nat(size_size_list_bool,L)))
<=> ( L != nil_bool ) ) ).
% length_ge_1_conv
tff(fact_1939_foldr__length,axiom,
! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_bi,L),zero_zero_nat) = aa_list_nat_nat(size_size_list_nat,L) ).
% foldr_length
tff(fact_1940_foldr__length,axiom,
! [L: list_int] : aa_nat_nat(foldr_int_nat(aTP_Lamm_bj,L),zero_zero_nat) = aa_list_int_nat(size_size_list_int,L) ).
% foldr_length
tff(fact_1941_foldr__length,axiom,
! [L: list_bool] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_bk,L),zero_zero_nat) = aa_list_bool_nat(size_size_list_bool,L) ).
% foldr_length
tff(fact_1942_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).
% le_ceiling_iff
tff(fact_1943_diff__self__eq__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).
% diff_self_eq_0
tff(fact_1944_diff__0__eq__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).
% diff_0_eq_0
tff(fact_1945_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
tff(fact_1946_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
tff(fact_1947_mult__0__right,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).
% mult_0_right
tff(fact_1948_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
tff(fact_1949_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).
% Suc_diff_diff
tff(fact_1950_diff__Suc__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).
% diff_Suc_Suc
tff(fact_1951_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).
% diff_diff_cancel
tff(fact_1952_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_1953_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
tff(fact_1954_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
tff(fact_1955_mult__zero__left,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).
% mult_zero_left
tff(fact_1956_mult__zero__left,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).
% mult_zero_left
tff(fact_1957_mult__zero__left,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).
% mult_zero_left
tff(fact_1958_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_1959_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_1960_mult__zero__right,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).
% mult_zero_right
tff(fact_1961_mult__zero__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% mult_zero_right
tff(fact_1962_mult__zero__right,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).
% mult_zero_right
tff(fact_1963_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_1964_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_1965_mult__eq__0__iff,axiom,
! [A: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = zero_z2226904508553997617atural )
<=> ( ( A = zero_z2226904508553997617atural )
| ( B = zero_z2226904508553997617atural ) ) ) ).
% mult_eq_0_iff
tff(fact_1966_mult__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
<=> ( ( A = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% mult_eq_0_iff
tff(fact_1967_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
tff(fact_1968_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_1969_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_1970_mult__cancel__left,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B) )
<=> ( ( C = zero_z2226904508553997617atural )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1971_mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1972_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1973_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_1974_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_1975_mult__cancel__right,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C) )
<=> ( ( C = zero_z2226904508553997617atural )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1976_mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1977_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1978_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_1979_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_1980_diff__self,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),A) = zero_z3403309356797280102nteger ).
% diff_self
tff(fact_1981_diff__self,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).
% diff_self
tff(fact_1982_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_1983_diff__0__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),zero_z3403309356797280102nteger) = A ).
% diff_0_right
tff(fact_1984_diff__0__right,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).
% diff_0_right
tff(fact_1985_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_1986_zero__diff,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).
% zero_diff
tff(fact_1987_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_1988_diff__zero,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),zero_z2226904508553997617atural) = A ).
% diff_zero
tff(fact_1989_diff__zero,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),zero_z3403309356797280102nteger) = A ).
% diff_zero
tff(fact_1990_diff__zero,axiom,
! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).
% diff_zero
tff(fact_1991_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_1992_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_1993_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),A) = zero_z2226904508553997617atural ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1994_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),A) = zero_z3403309356797280102nteger ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1995_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_1996_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_1997_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_1998_mult__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural),A) = A ).
% mult_1
tff(fact_1999_mult__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).
% mult_1
tff(fact_2000_mult__1,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).
% mult_1
tff(fact_2001_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_2002_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_2003_mult_Oright__neutral,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),one_one_Code_natural) = A ).
% mult.right_neutral
tff(fact_2004_mult_Oright__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).
% mult.right_neutral
tff(fact_2005_mult_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).
% mult.right_neutral
tff(fact_2006_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_2007_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_2008_add__diff__cancel__right_H,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_2009_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_2010_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_2011_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_2012_add__diff__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) ).
% add_diff_cancel_right
tff(fact_2013_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).
% add_diff_cancel_right
tff(fact_2014_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_2015_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_2016_add__diff__cancel__left_H,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_2017_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_2018_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_2019_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_2020_add__diff__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) ).
% add_diff_cancel_left
tff(fact_2021_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).
% add_diff_cancel_left
tff(fact_2022_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_2023_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_2024_diff__add__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),B) = A ).
% diff_add_cancel
tff(fact_2025_diff__add__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),B) = A ).
% diff_add_cancel
tff(fact_2026_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_2027_add__diff__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = A ).
% add_diff_cancel
tff(fact_2028_add__diff__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).
% add_diff_cancel
tff(fact_2029_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_2030_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_2031_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A) ).
% minus_diff_eq
tff(fact_2032_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_2033_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
<=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% one_eq_mult_iff
tff(fact_2034_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% mult_eq_1_iff
tff(fact_2035_zero__less__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% zero_less_diff
tff(fact_2036_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% nat_0_less_mult_iff
tff(fact_2037_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% mult_less_cancel2
tff(fact_2038_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_mult
tff(fact_2039_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_mult
tff(fact_2040_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% of_nat_mult
tff(fact_2041_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% of_nat_mult
tff(fact_2042_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% of_nat_mult
tff(fact_2043_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% diff_is_0_eq
tff(fact_2044_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).
% diff_is_0_eq'
tff(fact_2045_mult__Suc__right,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).
% mult_Suc_right
tff(fact_2046_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).
% Nat.add_diff_assoc
tff(fact_2047_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).
% Nat.add_diff_assoc2
tff(fact_2048_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).
% Nat.diff_diff_right
tff(fact_2049_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).
% of_int_mult
tff(fact_2050_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(aa_rat_fun_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_2051_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_2052_diff__Suc__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).
% diff_Suc_1
tff(fact_2053_of__int__diff,axiom,
! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).
% of_int_diff
tff(fact_2054_of__int__diff,axiom,
! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).
% of_int_diff
tff(fact_2055_of__int__diff,axiom,
! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).
% of_int_diff
tff(fact_2056_triangle__0,axiom,
aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).
% triangle_0
tff(fact_2057_diff__ge__0__iff__ge,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).
% diff_ge_0_iff_ge
tff(fact_2058_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% diff_ge_0_iff_ge
tff(fact_2059_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% diff_ge_0_iff_ge
tff(fact_2060_zero__comp__diff__simps_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).
% zero_comp_diff_simps(1)
tff(fact_2061_zero__comp__diff__simps_I1_J,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% zero_comp_diff_simps(1)
tff(fact_2062_zero__comp__diff__simps_I1_J,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% zero_comp_diff_simps(1)
tff(fact_2063_diff__gt__0__iff__gt,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ).
% diff_gt_0_iff_gt
tff(fact_2064_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).
% diff_gt_0_iff_gt
tff(fact_2065_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).
% diff_gt_0_iff_gt
tff(fact_2066_zero__comp__diff__simps_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ).
% zero_comp_diff_simps(2)
tff(fact_2067_zero__comp__diff__simps_I2_J,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).
% zero_comp_diff_simps(2)
tff(fact_2068_zero__comp__diff__simps_I2_J,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).
% zero_comp_diff_simps(2)
tff(fact_2069_le__add__diff__inverse,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_2070_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_2071_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_2072_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_2073_le__add__diff__inverse2,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_2074_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_2075_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_2076_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_2077_mult__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = C )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = one_one_Code_integer ) ) ) ).
% mult_cancel_right2
tff(fact_2078_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
<=> ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
tff(fact_2079_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_2080_mult__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_right1
tff(fact_2081_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
<=> ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
tff(fact_2082_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_2083_mult__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = C )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = one_one_Code_integer ) ) ) ).
% mult_cancel_left2
tff(fact_2084_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
<=> ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
tff(fact_2085_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_2086_mult__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_left1
tff(fact_2087_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
<=> ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
tff(fact_2088_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_2089_diff__add__zero,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) = zero_z2226904508553997617atural ).
% diff_add_zero
tff(fact_2090_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_2091_diff__numeral__special_I9_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% diff_numeral_special(9)
tff(fact_2092_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_2093_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_2094_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_2095_diff__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% diff_0
tff(fact_2096_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_2097_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_2098_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% verit_minus_simplify(3)
tff(fact_2099_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_2100_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_2101_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_2102_mult__minus1__right,axiom,
! [Z: rat] : aa_rat_rat(aa_rat_fun_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_2103_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_2104_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_2105_mult__minus1,axiom,
! [Z: rat] : aa_rat_rat(aa_rat_fun_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_2106_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_2107_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A) ).
% uminus_add_conv_diff
tff(fact_2108_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_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_2109_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_2110_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) ).
% diff_minus_eq_add
tff(fact_2111_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).
% diff_minus_eq_add
tff(fact_2112_Suc__pred,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).
% Suc_pred
tff(fact_2113_length__0__conv,axiom,
! [Xs: list_char] :
( ( size_size_list_char(Xs) = zero_zero_nat )
<=> ( Xs = nil_char ) ) ).
% length_0_conv
tff(fact_2114_length__0__conv,axiom,
! [Xs: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = zero_zero_nat )
<=> ( Xs = nil_nat ) ) ).
% length_0_conv
tff(fact_2115_length__0__conv,axiom,
! [Xs: list_int] :
( ( aa_list_int_nat(size_size_list_int,Xs) = zero_zero_nat )
<=> ( Xs = nil_int ) ) ).
% length_0_conv
tff(fact_2116_length__0__conv,axiom,
! [Xs: list_bool] :
( ( aa_list_bool_nat(size_size_list_bool,Xs) = zero_zero_nat )
<=> ( Xs = nil_bool ) ) ).
% length_0_conv
tff(fact_2117_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).
% one_le_mult_iff
tff(fact_2118_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% mult_le_cancel2
tff(fact_2119_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).
% diff_Suc_diff_eq1
tff(fact_2120_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).
% diff_Suc_diff_eq2
tff(fact_2121_Suc__diff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).
% Suc_diff
tff(fact_2122_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% zle_diff1_eq
tff(fact_2123_floor__diff__of__int,axiom,
! [X: rat,Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) ).
% floor_diff_of_int
tff(fact_2124_ceiling__diff__of__int,axiom,
! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z) ).
% ceiling_diff_of_int
tff(fact_2125_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_2126_diff__numeral__special_I12_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% diff_numeral_special(12)
tff(fact_2127_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_2128_length__greater__0__conv,axiom,
! [Xs: list_char] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_char(Xs)))
<=> ( Xs != nil_char ) ) ).
% length_greater_0_conv
tff(fact_2129_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)))
<=> ( Xs != nil_nat ) ) ).
% length_greater_0_conv
tff(fact_2130_length__greater__0__conv,axiom,
! [Xs: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs)))
<=> ( Xs != nil_int ) ) ).
% length_greater_0_conv
tff(fact_2131_length__greater__0__conv,axiom,
! [Xs: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,Xs)))
<=> ( Xs != nil_bool ) ) ).
% length_greater_0_conv
tff(fact_2132_Suc__diff__1,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).
% Suc_diff_1
tff(fact_2133_power__int__minus__one__mult__self,axiom,
! [M: int] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)) = one_one_rat ).
% power_int_minus_one_mult_self
tff(fact_2134_power__int__minus__one__mult__self_H,axiom,
! [M: int,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),B)) = B ).
% power_int_minus_one_mult_self'
tff(fact_2135_floor__diff__one,axiom,
! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ).
% floor_diff_one
tff(fact_2136_ceiling__diff__one,axiom,
! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).
% ceiling_diff_one
tff(fact_2137_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_2138_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: fun_nat_fun_nat_bool,Less: fun_nat_fun_nat_bool,Top: nat,A: nat] :
( pp(aa_nat_bool(ordering_top_nat(Less_eq,Less),Top))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(Less_eq,Top),A))
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
tff(fact_2139_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: fun_nat_fun_nat_bool,Less: fun_nat_fun_nat_bool,Top: nat,A: nat] :
( pp(aa_nat_bool(ordering_top_nat(Less_eq,Less),Top))
=> ( ( A != Top )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(Less,A),Top)) ) ) ).
% ordering_top.not_eq_extremum
tff(fact_2140_ordering__top_Oextremum__unique,axiom,
! [Less_eq: fun_nat_fun_nat_bool,Less: fun_nat_fun_nat_bool,Top: nat,A: nat] :
( pp(aa_nat_bool(ordering_top_nat(Less_eq,Less),Top))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(Less_eq,Top),A))
<=> ( A = Top ) ) ) ).
% ordering_top.extremum_unique
tff(fact_2141_ordering__top_Oextremum__strict,axiom,
! [Less_eq: fun_nat_fun_nat_bool,Less: fun_nat_fun_nat_bool,Top: nat,A: nat] :
( pp(aa_nat_bool(ordering_top_nat(Less_eq,Less),Top))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(Less,Top),A)) ) ).
% ordering_top.extremum_strict
tff(fact_2142_ordering__top_Oextremum,axiom,
! [Less_eq: fun_nat_fun_nat_bool,Less: fun_nat_fun_nat_bool,Top: nat,A: nat] :
( pp(aa_nat_bool(ordering_top_nat(Less_eq,Less),Top))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(Less_eq,A),Top)) ) ).
% ordering_top.extremum
tff(fact_2143_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2144_cancel__ab__semigroup__add__class_Odiff__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) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2145_cancel__ab__semigroup__add__class_Odiff__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) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2146_cancel__ab__semigroup__add__class_Odiff__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) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2147_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_2148_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_2149_ab__semigroup__mult__class_Omult_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)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_2150_ab__semigroup__mult__class_Omult_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)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_2151_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_2152_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_2153_ab__semigroup__mult__class_Omult_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) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_2154_ab__semigroup__mult__class_Omult_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) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_2155_mult_Oassoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% mult.assoc
tff(fact_2156_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% mult.assoc
tff(fact_2157_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_2158_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_2159_diff__eq__diff__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D) )
=> ( ( A = B )
<=> ( C = D ) ) ) ).
% diff_eq_diff_eq
tff(fact_2160_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_2161_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_2162_inf__period_I1_J,axiom,
! [P: fun_Co8822979462182099027r_bool,D3: code_integer,Q: fun_Co8822979462182099027r_bool] :
( ! [X2: code_integer,K3: code_integer] :
( pp(aa_Code_integer_bool(P,X2))
<=> pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))) )
=> ( ! [X2: code_integer,K3: code_integer] :
( pp(aa_Code_integer_bool(Q,X2))
<=> pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))) )
=> ! [X3: code_integer,K4: code_integer] :
( ( pp(aa_Code_integer_bool(P,X3))
& pp(aa_Code_integer_bool(Q,X3)) )
<=> ( pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3))))
& pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)))) ) ) ) ) ).
% inf_period(1)
tff(fact_2163_inf__period_I1_J,axiom,
! [P: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
( ! [X2: rat,K3: rat] :
( pp(aa_rat_bool(P,X2))
<=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))) )
=> ( ! [X2: rat,K3: rat] :
( pp(aa_rat_bool(Q,X2))
<=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))) )
=> ! [X3: rat,K4: rat] :
( ( pp(aa_rat_bool(P,X3))
& pp(aa_rat_bool(Q,X3)) )
<=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))))
& pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))) ) ) ) ) ).
% inf_period(1)
tff(fact_2164_inf__period_I1_J,axiom,
! [P: fun_int_bool,D3: int,Q: fun_int_bool] :
( ! [X2: int,K3: int] :
( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(Q,X2))
<=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ! [X3: int,K4: int] :
( ( pp(aa_int_bool(P,X3))
& pp(aa_int_bool(Q,X3)) )
<=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))))
& pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))) ) ) ) ) ).
% inf_period(1)
tff(fact_2165_inf__period_I2_J,axiom,
! [P: fun_Co8822979462182099027r_bool,D3: code_integer,Q: fun_Co8822979462182099027r_bool] :
( ! [X2: code_integer,K3: code_integer] :
( pp(aa_Code_integer_bool(P,X2))
<=> pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))) )
=> ( ! [X2: code_integer,K3: code_integer] :
( pp(aa_Code_integer_bool(Q,X2))
<=> pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))) )
=> ! [X3: code_integer,K4: code_integer] :
( ( pp(aa_Code_integer_bool(P,X3))
| pp(aa_Code_integer_bool(Q,X3)) )
<=> ( pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3))))
| pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)))) ) ) ) ) ).
% inf_period(2)
tff(fact_2166_inf__period_I2_J,axiom,
! [P: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
( ! [X2: rat,K3: rat] :
( pp(aa_rat_bool(P,X2))
<=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))) )
=> ( ! [X2: rat,K3: rat] :
( pp(aa_rat_bool(Q,X2))
<=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))) )
=> ! [X3: rat,K4: rat] :
( ( pp(aa_rat_bool(P,X3))
| pp(aa_rat_bool(Q,X3)) )
<=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))))
| pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))) ) ) ) ) ).
% inf_period(2)
tff(fact_2167_inf__period_I2_J,axiom,
! [P: fun_int_bool,D3: int,Q: fun_int_bool] :
( ! [X2: int,K3: int] :
( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(Q,X2))
<=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ! [X3: int,K4: int] :
( ( pp(aa_int_bool(P,X3))
| pp(aa_int_bool(Q,X3)) )
<=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))))
| pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))) ) ) ) ) ).
% inf_period(2)
tff(fact_2168_square__diff__square__factored,axiom,
! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Y)) ).
% square_diff_square_factored
tff(fact_2169_square__diff__square__factored,axiom,
! [X: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(minus_minus_rat(X),Y)) ).
% square_diff_square_factored
tff(fact_2170_square__diff__square__factored,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).
% square_diff_square_factored
tff(fact_2171_eq__add__iff2,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D) )
<=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)),E)),D) ) ) ).
% eq_add_iff2
tff(fact_2172_eq__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
<=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E)),D) ) ) ).
% eq_add_iff2
tff(fact_2173_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_2174_eq__add__iff1,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),E)),C) = D ) ) ).
% eq_add_iff1
tff(fact_2175_eq__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E)),C) = D ) ) ).
% eq_add_iff1
tff(fact_2176_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_2177_le__add__iff1,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),E)),C)),D)) ) ).
% le_add_iff1
tff(fact_2178_le__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D)) ) ).
% le_add_iff1
tff(fact_2179_le__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D)) ) ).
% le_add_iff1
tff(fact_2180_le__add__iff2,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)),E)),D))) ) ).
% le_add_iff2
tff(fact_2181_le__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E)),D))) ) ).
% le_add_iff2
tff(fact_2182_le__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D))) ) ).
% le_add_iff2
tff(fact_2183_less__add__iff2,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)),E)),D))) ) ).
% less_add_iff2
tff(fact_2184_less__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E)),D))) ) ).
% less_add_iff2
tff(fact_2185_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D))) ) ).
% less_add_iff2
tff(fact_2186_less__add__iff1,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),E)),C)),D)) ) ).
% less_add_iff1
tff(fact_2187_less__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D)) ) ).
% less_add_iff1
tff(fact_2188_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D)) ) ).
% less_add_iff1
tff(fact_2189_square__diff__one__factored,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),one_one_Code_integer)) ).
% square_diff_one_factored
tff(fact_2190_square__diff__one__factored,axiom,
! [X: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ).
% square_diff_one_factored
tff(fact_2191_square__diff__one__factored,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).
% square_diff_one_factored
tff(fact_2192_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).
% of_nat_diff
tff(fact_2193_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).
% of_nat_diff
tff(fact_2194_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ) ).
% of_nat_diff
tff(fact_2195_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).
% of_nat_diff
tff(fact_2196_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ) ) ).
% of_nat_diff
tff(fact_2197_minusinfinity,axiom,
! [D: int,P1: fun_int_bool,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(P1,X2))
<=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P1,X2)) ) )
=> ( ? [X_1: int] : pp(aa_int_bool(P1,X_1))
=> ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).
% minusinfinity
tff(fact_2198_plusinfinity,axiom,
! [D: int,P2: fun_int_bool,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(P2,X2))
<=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ? [X_1: int] : pp(aa_int_bool(P2,X_1))
=> ? [X_12: int] : pp(aa_int_bool(P,X_12)) ) ) ) ) ).
% plusinfinity
tff(fact_2199_int__minus,axiom,
! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).
% int_minus
tff(fact_2200_diff__eq__diff__less__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D)) ) ) ).
% diff_eq_diff_less_eq
tff(fact_2201_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) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).
% diff_eq_diff_less_eq
tff(fact_2202_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).
% diff_eq_diff_less_eq
tff(fact_2203_diff__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C))) ) ).
% diff_right_mono
tff(fact_2204_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).
% diff_right_mono
tff(fact_2205_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).
% diff_right_mono
tff(fact_2206_diff__left__mono,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B))) ) ).
% diff_left_mono
tff(fact_2207_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).
% diff_left_mono
tff(fact_2208_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).
% diff_left_mono
tff(fact_2209_diff__mono,axiom,
! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,D),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D))) ) ) ).
% diff_mono
tff(fact_2210_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).
% diff_mono
tff(fact_2211_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).
% diff_mono
tff(fact_2212_eq__iff__diff__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = zero_z3403309356797280102nteger ) ) ).
% eq_iff_diff_eq_0
tff(fact_2213_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_2214_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_2215_diff__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C))) ) ).
% diff_strict_right_mono
tff(fact_2216_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).
% diff_strict_right_mono
tff(fact_2217_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).
% diff_strict_right_mono
tff(fact_2218_diff__strict__left__mono,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B))) ) ).
% diff_strict_left_mono
tff(fact_2219_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).
% diff_strict_left_mono
tff(fact_2220_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).
% diff_strict_left_mono
tff(fact_2221_diff__eq__diff__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D)) ) ) ).
% diff_eq_diff_less
tff(fact_2222_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) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).
% diff_eq_diff_less
tff(fact_2223_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).
% diff_eq_diff_less
tff(fact_2224_diff__strict__mono,axiom,
! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,D),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D))) ) ) ).
% diff_strict_mono
tff(fact_2225_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).
% diff_strict_mono
tff(fact_2226_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).
% diff_strict_mono
tff(fact_2227_diff__diff__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% diff_diff_eq
tff(fact_2228_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(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% diff_diff_eq
tff(fact_2229_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_2230_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_2231_add__implies__diff,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) = A )
=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) ) ) ).
% add_implies_diff
tff(fact_2232_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
=> ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).
% add_implies_diff
tff(fact_2233_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_2234_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_2235_diff__add__eq__diff__diff__swap,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C)),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_2236_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_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_2237_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_2238_diff__add__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).
% diff_add_eq
tff(fact_2239_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).
% diff_add_eq
tff(fact_2240_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_2241_diff__diff__eq2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).
% diff_diff_eq2
tff(fact_2242_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(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).
% diff_diff_eq2
tff(fact_2243_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_2244_add__diff__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).
% add_diff_eq
tff(fact_2245_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).
% add_diff_eq
tff(fact_2246_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_2247_eq__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_2248_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A = aa_rat_rat(minus_minus_rat(C),B) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_2249_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_2250_diff__eq__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = C )
<=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) ) ) ).
% diff_eq_eq
tff(fact_2251_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(minus_minus_rat(A),B) = C )
<=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).
% diff_eq_eq
tff(fact_2252_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_2253_group__cancel_Osub1,axiom,
! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
( ( A3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A3),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ) ) ).
% group_cancel.sub1
tff(fact_2254_group__cancel_Osub1,axiom,
! [A3: rat,K: rat,A: rat,B: rat] :
( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
=> ( aa_rat_rat(minus_minus_rat(A3),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).
% group_cancel.sub1
tff(fact_2255_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_2256_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_2257_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ).
% minus_diff_commute
tff(fact_2258_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_2259_mult__not__zero,axiom,
! [A: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) != zero_z2226904508553997617atural )
=> ( ( A != zero_z2226904508553997617atural )
& ( B != zero_z2226904508553997617atural ) ) ) ).
% mult_not_zero
tff(fact_2260_mult__not__zero,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger )
=> ( ( A != zero_z3403309356797280102nteger )
& ( B != zero_z3403309356797280102nteger ) ) ) ).
% mult_not_zero
tff(fact_2261_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
tff(fact_2262_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_2263_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_2264_divisors__zero,axiom,
! [A: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = zero_z2226904508553997617atural )
=> ( ( A = zero_z2226904508553997617atural )
| ( B = zero_z2226904508553997617atural ) ) ) ).
% divisors_zero
tff(fact_2265_divisors__zero,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
=> ( ( A = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% divisors_zero
tff(fact_2266_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
tff(fact_2267_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
tff(fact_2268_divisors__zero,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
tff(fact_2269_no__zero__divisors,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) != zero_z2226904508553997617atural ) ) ) ).
% no_zero_divisors
tff(fact_2270_no__zero__divisors,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger ) ) ) ).
% no_zero_divisors
tff(fact_2271_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).
% no_zero_divisors
tff(fact_2272_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_2273_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_2274_mult__left__cancel,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2275_mult__left__cancel,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2276_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2277_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_2278_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_2279_mult__right__cancel,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2280_mult__right__cancel,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2281_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2282_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_2283_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_2284_crossproduct__eq,axiom,
! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_2285_crossproduct__eq,axiom,
! [W: rat,Y: rat,X: rat,Z: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_2286_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_2287_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_2288_crossproduct__noteq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( ( A != B )
& ( C != D ) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_2289_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( A != B )
& ( C != D ) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_2290_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_2291_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_2292_ring__class_Oring__distribs_I2_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_2293_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_2294_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_2295_ring__class_Oring__distribs_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_2296_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_2297_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_2298_comm__semiring__class_Odistrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% comm_semiring_class.distrib
tff(fact_2299_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% comm_semiring_class.distrib
tff(fact_2300_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_2301_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_2302_distrib__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).
% distrib_left
tff(fact_2303_distrib__left,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).
% distrib_left
tff(fact_2304_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_2305_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_2306_distrib__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% distrib_right
tff(fact_2307_distrib__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% distrib_right
tff(fact_2308_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_2309_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_2310_combine__common__factor,axiom,
! [A: code_integer,E: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),E)),C) ).
% combine_common_factor
tff(fact_2311_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C) ).
% combine_common_factor
tff(fact_2312_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_2313_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_2314_mult_Ocomm__neutral,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),one_one_Code_natural) = A ).
% mult.comm_neutral
tff(fact_2315_mult_Ocomm__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).
% mult.comm_neutral
tff(fact_2316_mult_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).
% mult.comm_neutral
tff(fact_2317_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_2318_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_2319_comm__monoid__mult__class_Omult__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_2320_comm__monoid__mult__class_Omult__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_2321_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_2322_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_2323_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_2324_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
tff(fact_2325_minus__nat_Odiff__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).
% minus_nat.diff_0
tff(fact_2326_minus__int__code_I1_J,axiom,
! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).
% minus_int_code(1)
tff(fact_2327_zero__induct__lemma,axiom,
! [P: fun_nat_bool,K: nat,I: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).
% zero_induct_lemma
tff(fact_2328_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).
% diff_less_mono2
tff(fact_2329_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).
% less_imp_diff_less
tff(fact_2330_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).
% diff_le_mono2
tff(fact_2331_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).
% le_diff_iff'
tff(fact_2332_diff__le__self,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).
% diff_le_self
tff(fact_2333_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).
% diff_le_mono
tff(fact_2334_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).
% Nat.diff_diff_eq
tff(fact_2335_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).
% le_diff_iff
tff(fact_2336_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
<=> ( M = N ) ) ) ) ).
% eq_diff_iff
tff(fact_2337_diff__add__inverse2,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).
% diff_add_inverse2
tff(fact_2338_diff__add__inverse,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).
% diff_add_inverse
tff(fact_2339_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).
% diff_cancel2
tff(fact_2340_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).
% Nat.diff_cancel
tff(fact_2341_mult__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).
% mult_0
tff(fact_2342_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
<=> ( M = N ) ) ).
% Suc_mult_cancel1
tff(fact_2343_mult__of__nat__commute,axiom,
! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).
% mult_of_nat_commute
tff(fact_2344_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).
% mult_of_nat_commute
tff(fact_2345_mult__of__nat__commute,axiom,
! [X: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,X)) ).
% mult_of_nat_commute
tff(fact_2346_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).
% mult_of_nat_commute
tff(fact_2347_mult__of__nat__commute,axiom,
! [X: nat,Y: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),Y) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Y),aa_nat_Code_natural(semiri3763490453095760265atural,X)) ).
% mult_of_nat_commute
tff(fact_2348_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_2349_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_2350_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).
% mult_le_mono2
tff(fact_2351_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).
% mult_le_mono1
tff(fact_2352_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).
% mult_le_mono
tff(fact_2353_le__square,axiom,
! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).
% le_square
tff(fact_2354_le__cube,axiom,
! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).
% le_cube
tff(fact_2355_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).
% int_diff_cases
tff(fact_2356_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).
% add_mult_distrib2
tff(fact_2357_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).
% add_mult_distrib
tff(fact_2358_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_2359_nat__mult__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).
% nat_mult_1
tff(fact_2360_nat__mult__1__right,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).
% nat_mult_1_right
tff(fact_2361_mult__of__int__commute,axiom,
! [X: int,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).
% mult_of_int_commute
tff(fact_2362_mult__of__int__commute,axiom,
! [X: int,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_int_rat(ring_1_of_int_rat,X)) ).
% mult_of_int_commute
tff(fact_2363_mult__of__int__commute,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).
% mult_of_int_commute
tff(fact_2364_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_2365_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_2366_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_2367_power__int__mult__distrib,axiom,
! [X: rat,Y: rat,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(Y),M)) ).
% power_int_mult_distrib
tff(fact_2368_power__int__commutes,axiom,
! [X: rat,N: int] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),N)),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_int_rat(power_int_rat(X),N)) ).
% power_int_commutes
tff(fact_2369_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).
% int_ops(6)
tff(fact_2370_mult__eq__if,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).
% mult_eq_if
tff(fact_2371_decr__mult__lemma,axiom,
! [D: int,P: fun_int_bool,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X2: int] :
( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D))) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X3: int] :
( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).
% decr_mult_lemma
tff(fact_2372_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).
% nat_diff_distrib'
tff(fact_2373_nat__diff__distrib,axiom,
! [Z4: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z4),Z))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).
% nat_diff_distrib
tff(fact_2374_lambda__zero,axiom,
aTP_Lamm_bl = aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_z2226904508553997617atural) ).
% lambda_zero
tff(fact_2375_lambda__zero,axiom,
aTP_Lamm_bm = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger) ).
% lambda_zero
tff(fact_2376_lambda__zero,axiom,
aTP_Lamm_bn = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).
% lambda_zero
tff(fact_2377_lambda__zero,axiom,
aTP_Lamm_bo = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).
% lambda_zero
tff(fact_2378_lambda__zero,axiom,
aTP_Lamm_bp = aa_int_fun_int_int(times_times_int,zero_zero_int) ).
% lambda_zero
tff(fact_2379_lambda__one,axiom,
aTP_Lamm_bq = aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural) ).
% lambda_one
tff(fact_2380_lambda__one,axiom,
aTP_Lamm_br = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).
% lambda_one
tff(fact_2381_lambda__one,axiom,
aTP_Lamm_bs = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).
% lambda_one
tff(fact_2382_lambda__one,axiom,
aTP_Lamm_bt = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).
% lambda_one
tff(fact_2383_lambda__one,axiom,
aTP_Lamm_bu = aa_int_fun_int_int(times_times_int,one_one_int) ).
% lambda_one
tff(fact_2384_nat__mult__distrib,axiom,
! [Z: int,Z4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).
% nat_mult_distrib
tff(fact_2385_zdiff__int__split,axiom,
! [P: fun_int_bool,X: nat,Y: nat] :
( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).
% zdiff_int_split
tff(fact_2386_le__iff__diff__le__0,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),zero_z3403309356797280102nteger)) ) ).
% le_iff_diff_le_0
tff(fact_2387_le__iff__diff__le__0,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).
% le_iff_diff_le_0
tff(fact_2388_le__iff__diff__le__0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).
% le_iff_diff_le_0
tff(fact_2389_power__int__minus__mult,axiom,
! [X: rat,N: int] :
( ( ( X != zero_zero_rat )
| ( N != zero_zero_int ) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int))),X) = aa_int_rat(power_int_rat(X),N) ) ) ).
% power_int_minus_mult
tff(fact_2390_less__iff__diff__less__0,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),zero_z3403309356797280102nteger)) ) ).
% less_iff_diff_less_0
tff(fact_2391_less__iff__diff__less__0,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).
% less_iff_diff_less_0
tff(fact_2392_less__iff__diff__less__0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).
% less_iff_diff_less_0
tff(fact_2393_le__mult__floor,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).
% le_mult_floor
tff(fact_2394_diff__le__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))) ) ).
% diff_le_eq
tff(fact_2395_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).
% diff_le_eq
tff(fact_2396_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% diff_le_eq
tff(fact_2397_le__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C)) ) ).
% le_diff_eq
tff(fact_2398_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).
% le_diff_eq
tff(fact_2399_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).
% le_diff_eq
tff(fact_2400_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_2401_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).
% le_add_diff
tff(fact_2402_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B)) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_2403_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_2404_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_2405_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_2406_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2407_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2408_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_2409_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
<=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2410_add__le__imp__le__diff,axiom,
! [I: code_integer,K: code_integer,N: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,N),K))) ) ).
% add_le_imp_le_diff
tff(fact_2411_add__le__imp__le__diff,axiom,
! [I: rat,K: rat,N: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(N),K))) ) ).
% add_le_imp_le_diff
tff(fact_2412_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).
% add_le_imp_le_diff
tff(fact_2413_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).
% add_le_imp_le_diff
tff(fact_2414_add__le__add__imp__diff__le,axiom,
! [I: code_integer,K: code_integer,N: code_integer,J: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),N))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),K)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),N))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),K)))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_2415_add__le__add__imp__diff__le,axiom,
! [I: rat,K: rat,N: rat,J: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_2416_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_2417_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_2418_mult__ceiling__le,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_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_2419_less__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C)) ) ).
% less_diff_eq
tff(fact_2420_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).
% less_diff_eq
tff(fact_2421_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).
% less_diff_eq
tff(fact_2422_diff__less__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),C))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))) ) ).
% diff_less_eq
tff(fact_2423_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).
% diff_less_eq
tff(fact_2424_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% diff_less_eq
tff(fact_2425_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: code_integer,B: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_2426_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_2427_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_2428_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_2429_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_2430_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2431_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(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2432_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_2433_diff__conv__add__uminus,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% diff_conv_add_uminus
tff(fact_2434_diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).
% diff_conv_add_uminus
tff(fact_2435_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_2436_group__cancel_Osub2,axiom,
! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ) ) ).
% group_cancel.sub2
tff(fact_2437_group__cancel_Osub2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
=> ( aa_rat_rat(minus_minus_rat(A),B3) = aa_rat_rat(aa_rat_fun_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_2438_nat__mult__distrib__neg,axiom,
! [Z: int,Z4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z4))) ) ) ).
% nat_mult_distrib_neg
tff(fact_2439_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2440_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2441_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2442_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2443_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2444_zero__le__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) ) ) ).
% zero_le_mult_iff
tff(fact_2445_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).
% zero_le_mult_iff
tff(fact_2446_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).
% zero_le_mult_iff
tff(fact_2447_mult__nonneg__nonpos2,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)),zero_z2226904508553997617atural)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2448_mult__nonneg__nonpos2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2449_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2450_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2451_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2452_mult__nonpos__nonneg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2453_mult__nonpos__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2454_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2455_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2456_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2457_mult__nonneg__nonpos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2458_mult__nonneg__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2459_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2460_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2461_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2462_mult__nonneg__nonneg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2463_mult__nonneg__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2464_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2465_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2466_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2467_split__mult__neg__le,axiom,
! [A: code_natural,B: code_natural] :
( ( ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
& pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural)) )
| ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural))
& pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)) ) )
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural)) ) ).
% split_mult_neg_le
tff(fact_2468_split__mult__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)) ) ).
% split_mult_neg_le
tff(fact_2469_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ).
% split_mult_neg_le
tff(fact_2470_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ).
% split_mult_neg_le
tff(fact_2471_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ).
% split_mult_neg_le
tff(fact_2472_mult__le__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) ) ) ).
% mult_le_0_iff
tff(fact_2473_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).
% mult_le_0_iff
tff(fact_2474_mult__le__0__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).
% mult_le_0_iff
tff(fact_2475_mult__right__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))) ) ) ).
% mult_right_mono
tff(fact_2476_mult__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ) ).
% mult_right_mono
tff(fact_2477_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).
% mult_right_mono
tff(fact_2478_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).
% mult_right_mono
tff(fact_2479_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_right_mono
tff(fact_2480_mult__right__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ) ).
% mult_right_mono_neg
tff(fact_2481_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).
% mult_right_mono_neg
tff(fact_2482_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_right_mono_neg
tff(fact_2483_mult__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))) ) ) ).
% mult_left_mono
tff(fact_2484_mult__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ).
% mult_left_mono
tff(fact_2485_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).
% mult_left_mono
tff(fact_2486_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% mult_left_mono
tff(fact_2487_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_left_mono
tff(fact_2488_mult__nonpos__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ) ).
% mult_nonpos_nonpos
tff(fact_2489_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).
% mult_nonpos_nonpos
tff(fact_2490_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_nonpos_nonpos
tff(fact_2491_mult__left__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ).
% mult_left_mono_neg
tff(fact_2492_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).
% mult_left_mono_neg
tff(fact_2493_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_left_mono_neg
tff(fact_2494_split__mult__pos__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ).
% split_mult_pos_le
tff(fact_2495_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ).
% split_mult_pos_le
tff(fact_2496_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).
% split_mult_pos_le
tff(fact_2497_zero__le__square,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A))) ).
% zero_le_square
tff(fact_2498_zero__le__square,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A))) ).
% zero_le_square
tff(fact_2499_zero__le__square,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A))) ).
% zero_le_square
tff(fact_2500_mult__mono_H,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,C),D))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2501_mult__mono_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2502_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2503_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2504_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2505_mult__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,C),D))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2506_mult__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2507_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2508_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2509_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2510_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2511_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2512_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2513_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2514_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2515_mult__less__cancel__right__disj,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2516_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2517_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2518_mult__strict__right__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2519_mult__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2520_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2521_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2522_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2523_mult__strict__right__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2524_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2525_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2526_mult__less__cancel__left__disj,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2527_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2528_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2529_mult__strict__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2530_mult__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2531_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2532_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2533_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2534_mult__strict__left__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2535_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2536_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2537_mult__less__cancel__left__pos,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2538_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2539_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2540_mult__less__cancel__left__neg,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2541_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2542_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2543_zero__less__mult__pos2,axiom,
! [B: code_natural,A: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2544_zero__less__mult__pos2,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2545_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2546_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2547_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2548_zero__less__mult__pos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2549_zero__less__mult__pos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2550_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2551_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2552_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2553_zero__less__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)) ) ) ) ).
% zero_less_mult_iff
tff(fact_2554_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).
% zero_less_mult_iff
tff(fact_2555_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).
% zero_less_mult_iff
tff(fact_2556_mult__pos__neg2,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)),zero_z2226904508553997617atural)) ) ) ).
% mult_pos_neg2
tff(fact_2557_mult__pos__neg2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger)) ) ) ).
% mult_pos_neg2
tff(fact_2558_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).
% mult_pos_neg2
tff(fact_2559_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).
% mult_pos_neg2
tff(fact_2560_mult__pos__neg2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).
% mult_pos_neg2
tff(fact_2561_mult__pos__pos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2562_mult__pos__pos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2563_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2564_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2565_mult__pos__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2566_mult__pos__neg,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% mult_pos_neg
tff(fact_2567_mult__pos__neg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% mult_pos_neg
tff(fact_2568_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).
% mult_pos_neg
tff(fact_2569_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_pos_neg
tff(fact_2570_mult__pos__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_pos_neg
tff(fact_2571_mult__neg__pos,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural)) ) ) ).
% mult_neg_pos
tff(fact_2572_mult__neg__pos,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)) ) ) ).
% mult_neg_pos
tff(fact_2573_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).
% mult_neg_pos
tff(fact_2574_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_neg_pos
tff(fact_2575_mult__neg__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_neg_pos
tff(fact_2576_mult__less__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)) )
| ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ) ) ).
% mult_less_0_iff
tff(fact_2577_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).
% mult_less_0_iff
tff(fact_2578_mult__less__0__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).
% mult_less_0_iff
tff(fact_2579_not__square__less__zero,axiom,
! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),zero_z3403309356797280102nteger)) ).
% not_square_less_zero
tff(fact_2580_not__square__less__zero,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat)) ).
% not_square_less_zero
tff(fact_2581_not__square__less__zero,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int)) ).
% not_square_less_zero
tff(fact_2582_mult__neg__neg,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ) ).
% mult_neg_neg
tff(fact_2583_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).
% mult_neg_neg
tff(fact_2584_mult__neg__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_neg_neg
tff(fact_2585_add__scale__eq__noteq,axiom,
! [R: code_integer,A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( R != zero_z3403309356797280102nteger )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),C)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_2586_add__scale__eq__noteq,axiom,
! [R: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_2587_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_2588_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_2589_less__1__mult,axiom,
! [M: code_natural,N: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),M))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),N))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,M),N))) ) ) ).
% less_1_mult
tff(fact_2590_less__1__mult,axiom,
! [M: code_integer,N: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),M))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M),N))) ) ) ).
% less_1_mult
tff(fact_2591_less__1__mult,axiom,
! [M: rat,N: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),N))) ) ) ).
% less_1_mult
tff(fact_2592_less__1__mult,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
% less_1_mult
tff(fact_2593_less__1__mult,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).
% less_1_mult
tff(fact_2594_Suc__to__right,axiom,
! [N: nat,M: nat] :
( ( aa_nat_nat(suc,N) = M )
=> ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).
% Suc_to_right
tff(fact_2595_diff__less,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).
% diff_less
tff(fact_2596_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).
% Suc_diff_Suc
tff(fact_2597_diff__less__Suc,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).
% diff_less_Suc
tff(fact_2598_square__eq__1__iff,axiom,
! [X: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
<=> ( ( X = one_one_int )
| ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% square_eq_1_iff
tff(fact_2599_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
<=> ( ( X = one_one_Code_integer )
| ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% square_eq_1_iff
tff(fact_2600_square__eq__1__iff,axiom,
! [X: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X) = one_one_rat )
<=> ( ( X = one_one_rat )
| ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).
% square_eq_1_iff
tff(fact_2601_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).
% Suc_diff_le
tff(fact_2602_diff__add__0,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).
% diff_add_0
tff(fact_2603_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).
% less_diff_iff
tff(fact_2604_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).
% diff_less_mono
tff(fact_2605_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).
% add_diff_inverse_nat
tff(fact_2606_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).
% less_diff_conv
tff(fact_2607_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).
% le_diff_conv
tff(fact_2608_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).
% Nat.le_diff_conv2
tff(fact_2609_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).
% Nat.diff_add_assoc
tff(fact_2610_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).
% Nat.diff_add_assoc2
tff(fact_2611_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
<=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).
% Nat.le_imp_diff_is_add
tff(fact_2612_minus__int__code_I2_J,axiom,
! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).
% minus_int_code(2)
tff(fact_2613_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).
% mult_less_mono2
tff(fact_2614_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).
% mult_less_mono1
tff(fact_2615_int__le__induct,axiom,
! [I: int,K: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
=> ( pp(aa_int_bool(P,K))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_le_induct
tff(fact_2616_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_mult_less_cancel1
tff(fact_2617_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).
% diff_Suc_eq_diff_pred
tff(fact_2618_diff__nat__eq__if,axiom,
! [Z4: int,Z: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),zero_zero_int))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = aa_int_nat(nat2,Z) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),zero_zero_int))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4))) ) ) ) ).
% diff_nat_eq_if
tff(fact_2619_int__less__induct,axiom,
! [I: int,K: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_less_induct
tff(fact_2620_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% Suc_mult_le_cancel1
tff(fact_2621_mult__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).
% mult_Suc
tff(fact_2622_mlex__snd__decrI,axiom,
! [A: nat,A4: nat,B: nat,B4: nat,N4: nat] :
( ( A = A4 )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),B4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N4)),B4))) ) ) ).
% mlex_snd_decrI
tff(fact_2623_mlex__fst__decrI,axiom,
! [A: nat,A4: nat,B: nat,N4: nat,B4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B4),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N4)),B4))) ) ) ) ).
% mlex_fst_decrI
tff(fact_2624_mlex__bound,axiom,
! [A: nat,A3: nat,B: nat,N4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N4))) ) ) ).
% mlex_bound
tff(fact_2625_mlex__leI,axiom,
! [A: nat,A4: nat,B: nat,B4: nat,N4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),B4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N4)),B4))) ) ) ).
% mlex_leI
tff(fact_2626_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
tff(fact_2627_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).
% zmult_zless_mono2
tff(fact_2628_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
<=> ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
& ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).
% zmult_eq_1_iff
tff(fact_2629_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
=> ( ( M = one_one_int )
| ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
tff(fact_2630_list_Osize_I3_J,axiom,
size_size_list_char(nil_char) = zero_zero_nat ).
% list.size(3)
tff(fact_2631_list_Osize_I3_J,axiom,
aa_list_nat_nat(size_size_list_nat,nil_nat) = zero_zero_nat ).
% list.size(3)
tff(fact_2632_list_Osize_I3_J,axiom,
aa_list_int_nat(size_size_list_int,nil_int) = zero_zero_nat ).
% list.size(3)
tff(fact_2633_list_Osize_I3_J,axiom,
aa_list_bool_nat(size_size_list_bool,nil_bool) = zero_zero_nat ).
% list.size(3)
tff(fact_2634_len__greater__imp__nonempty,axiom,
! [X: nat,L: list_char] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),size_size_list_char(L)))
=> ( L != nil_char ) ) ).
% len_greater_imp_nonempty
tff(fact_2635_len__greater__imp__nonempty,axiom,
! [X: nat,L: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_nat_nat(size_size_list_nat,L)))
=> ( L != nil_nat ) ) ).
% len_greater_imp_nonempty
tff(fact_2636_len__greater__imp__nonempty,axiom,
! [X: nat,L: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_int_nat(size_size_list_int,L)))
=> ( L != nil_int ) ) ).
% len_greater_imp_nonempty
tff(fact_2637_len__greater__imp__nonempty,axiom,
! [X: nat,L: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_bool_nat(size_size_list_bool,L)))
=> ( L != nil_bool ) ) ).
% len_greater_imp_nonempty
tff(fact_2638_mult__less__le__imp__less,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,C),D))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2639_mult__less__le__imp__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2640_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2641_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2642_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2643_mult__le__less__imp__less,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,C),D))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2644_mult__le__less__imp__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2645_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2646_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2647_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2648_mult__right__le__imp__le,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2649_mult__right__le__imp__le,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2650_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2651_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2652_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2653_mult__left__le__imp__le,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2654_mult__left__le__imp__le,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2655_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2656_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2657_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2658_mult__le__cancel__left__pos,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2659_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2660_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2661_mult__le__cancel__left__neg,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2662_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2663_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2664_mult__less__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).
% mult_less_cancel_right
tff(fact_2665_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_right
tff(fact_2666_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_right
tff(fact_2667_mult__strict__mono_H,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,C),D))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2668_mult__strict__mono_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2669_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2670_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2671_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2672_mult__right__less__imp__less,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2673_mult__right__less__imp__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2674_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2675_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2676_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2677_mult__less__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).
% mult_less_cancel_left
tff(fact_2678_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_left
tff(fact_2679_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_left
tff(fact_2680_mult__strict__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,C),D))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2681_mult__strict__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2682_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2683_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2684_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2685_mult__left__less__imp__less,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2686_mult__left__less__imp__less,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2687_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2688_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2689_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2690_mult__le__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ) ).
% mult_le_cancel_right
tff(fact_2691_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% mult_le_cancel_right
tff(fact_2692_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).
% mult_le_cancel_right
tff(fact_2693_mult__le__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ) ).
% mult_le_cancel_left
tff(fact_2694_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% mult_le_cancel_left
tff(fact_2695_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).
% mult_le_cancel_left
tff(fact_2696_sum__squares__ge__zero,axiom,
! [X: code_integer,Y: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)))) ).
% sum_squares_ge_zero
tff(fact_2697_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))) ).
% sum_squares_ge_zero
tff(fact_2698_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).
% sum_squares_ge_zero
tff(fact_2699_mult__left__le,axiom,
! [C: code_natural,A: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,C),one_one_Code_natural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2700_mult__left__le,axiom,
! [C: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2701_mult__left__le,axiom,
! [C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2702_mult__left__le,axiom,
! [C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2703_mult__left__le,axiom,
! [C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2704_mult__le__one,axiom,
! [A: code_natural,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),one_one_Code_natural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),one_one_Code_natural)) ) ) ) ).
% mult_le_one
tff(fact_2705_mult__le__one,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)) ) ) ) ).
% mult_le_one
tff(fact_2706_mult__le__one,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat)) ) ) ) ).
% mult_le_one
tff(fact_2707_mult__le__one,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ) ).
% mult_le_one
tff(fact_2708_mult__le__one,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ) ).
% mult_le_one
tff(fact_2709_mult__right__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X)) ) ) ) ).
% mult_right_le_one_le
tff(fact_2710_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X)) ) ) ) ).
% mult_right_le_one_le
tff(fact_2711_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).
% mult_right_le_one_le
tff(fact_2712_mult__left__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X)) ) ) ) ).
% mult_left_le_one_le
tff(fact_2713_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X)) ) ) ) ).
% mult_left_le_one_le
tff(fact_2714_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).
% mult_left_le_one_le
tff(fact_2715_not__sum__squares__lt__zero,axiom,
! [X: code_integer,Y: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger)) ).
% not_sum_squares_lt_zero
tff(fact_2716_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)) ).
% not_sum_squares_lt_zero
tff(fact_2717_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).
% not_sum_squares_lt_zero
tff(fact_2718_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).
% ex_less_of_nat_mult
tff(fact_2719_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).
% diff_Suc_less
tff(fact_2720_nat__diff__split__asm,axiom,
! [P: fun_nat_bool,A: nat,B: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
<=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
& ~ pp(aa_nat_bool(P,zero_zero_nat)) )
| ? [D4: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
& ~ pp(aa_nat_bool(P,D4)) ) ) ) ).
% nat_diff_split_asm
tff(fact_2721_nat__diff__split,axiom,
! [P: fun_nat_bool,A: nat,B: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(P,zero_zero_nat)) )
& ! [D4: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
=> pp(aa_nat_bool(P,D4)) ) ) ) ).
% nat_diff_split
tff(fact_2722_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).
% less_diff_conv2
tff(fact_2723_power__int__minus__one__diff__commute,axiom,
! [A: int,B: int] : aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).
% power_int_minus_one_diff_commute
tff(fact_2724_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).
% n_less_n_mult_m
tff(fact_2725_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
% n_less_m_mult_n
tff(fact_2726_one__less__mult,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
% one_less_mult
tff(fact_2727_int__induct,axiom,
! [P: fun_int_bool,K: int,I: int] :
( pp(aa_int_bool(P,K))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_induct
tff(fact_2728_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
tff(fact_2729_frac__def,axiom,
! [X: rat] : aa_rat_rat(archimedean_frac_rat,X) = aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))) ).
% frac_def
tff(fact_2730_dbl__dec__def,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).
% dbl_dec_def
tff(fact_2731_dbl__dec__def,axiom,
! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).
% dbl_dec_def
tff(fact_2732_dbl__dec__def,axiom,
! [X: int] : neg_nu3811975205180677377ec_int(X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).
% dbl_dec_def
tff(fact_2733_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),X)),Y)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% field_le_mult_one_interval
tff(fact_2734_mult__less__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A)) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2735_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2736_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2737_mult__less__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer)) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2738_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2739_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2740_mult__less__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A)) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2741_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2742_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2743_mult__less__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer)) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2744_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2745_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2746_mult__le__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A)) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2747_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2748_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2749_mult__le__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer)) ) ) ) ).
% mult_le_cancel_right1
tff(fact_2750_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).
% mult_le_cancel_right1
tff(fact_2751_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).
% mult_le_cancel_right1
tff(fact_2752_mult__le__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A)) ) ) ) ).
% mult_le_cancel_left2
tff(fact_2753_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).
% mult_le_cancel_left2
tff(fact_2754_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).
% mult_le_cancel_left2
tff(fact_2755_mult__le__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer)) ) ) ) ).
% mult_le_cancel_left1
tff(fact_2756_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).
% mult_le_cancel_left1
tff(fact_2757_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).
% mult_le_cancel_left1
tff(fact_2758_convex__bound__le,axiom,
! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_le
tff(fact_2759_convex__bound__le,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_le
tff(fact_2760_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_le
tff(fact_2761_nz__le__conv__less,axiom,
! [K: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M)) ) ) ).
% nz_le_conv_less
tff(fact_2762_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% Suc_diff_eq_diff_pred
tff(fact_2763_Suc__pred_H,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% Suc_pred'
tff(fact_2764_frac__add,axiom,
! [X: rat,Y: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
=> ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
=> ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).
% frac_add
tff(fact_2765_add__eq__if,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).
% add_eq_if
tff(fact_2766_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).
% Suc_n_minus_m_eq
tff(fact_2767_foldr__length__aux,axiom,
! [L: list_nat,A: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_bi,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_nat_nat(size_size_list_nat,L)) ).
% foldr_length_aux
tff(fact_2768_foldr__length__aux,axiom,
! [L: list_int,A: nat] : aa_nat_nat(foldr_int_nat(aTP_Lamm_bj,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_int_nat(size_size_list_int,L)) ).
% foldr_length_aux
tff(fact_2769_foldr__length__aux,axiom,
! [L: list_bool,A: nat] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_bk,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_bool_nat(size_size_list_bool,L)) ).
% foldr_length_aux
tff(fact_2770_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).
% zmult_zless_mono2_lemma
tff(fact_2771_power__int__add,axiom,
! [X: rat,M: int,N: int] :
( ( ( X != zero_zero_rat )
| ( aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N) != zero_zero_int ) )
=> ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)) ) ) ).
% power_int_add
tff(fact_2772_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R3: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ).
% unique_quotient_lemma_neg
tff(fact_2773_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R3: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R3),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ).
% unique_quotient_lemma
tff(fact_2774_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q2: int,R: int,B4: int,Q5: int,R3: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R3) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R3)),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
tff(fact_2775_zdiv__mono2__lemma,axiom,
! [B: int,Q2: int,R: int,B4: int,Q5: int,R3: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R3) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R3)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R3),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
tff(fact_2776_q__pos__lemma,axiom,
! [B4: int,Q5: int,R3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R3)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R3),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).
% q_pos_lemma
tff(fact_2777_incr__mult__lemma,axiom,
! [D: int,P: fun_int_bool,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X2: int] :
( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D))) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X3: int] :
( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).
% incr_mult_lemma
tff(fact_2778_ceiling__diff__floor__le__1,axiom,
! [X: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X))),one_one_int)) ).
% ceiling_diff_floor_le_1
tff(fact_2779_convex__bound__lt,axiom,
! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_2780_convex__bound__lt,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_2781_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_2782_ceiling__correct,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ) ).
% ceiling_correct
tff(fact_2783_ceiling__unique,axiom,
! [Z: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
=> ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).
% ceiling_unique
tff(fact_2784_ceiling__eq__iff,axiom,
! [X: rat,A: int] :
( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))) ) ) ).
% ceiling_eq_iff
tff(fact_2785_ceiling__split,axiom,
! [P: fun_int_bool,T: rat] :
( pp(aa_int_bool(P,aa_rat_int(archim2889992004027027881ng_rat,T)))
<=> ! [I3: int] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)),T))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I3))) )
=> pp(aa_int_bool(P,I3)) ) ) ).
% ceiling_split
tff(fact_2786_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).
% ceiling_less_iff
tff(fact_2787_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% nat_mult_le_cancel_disj
tff(fact_2788_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% nat_mult_less_cancel_disj
tff(fact_2789_sum__squares__gt__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))))
<=> ( ( X != zero_z3403309356797280102nteger )
| ( Y != zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2790_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))))
<=> ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2791_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
<=> ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2792_sum__squares__le__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger))
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2793_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat))
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2794_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2795_mult__le__cancel__iff2,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).
% mult_le_cancel_iff2
tff(fact_2796_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).
% mult_le_cancel_iff2
tff(fact_2797_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).
% mult_le_cancel_iff2
tff(fact_2798_mult__le__cancel__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).
% mult_le_cancel_iff1
tff(fact_2799_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).
% mult_le_cancel_iff1
tff(fact_2800_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).
% mult_le_cancel_iff1
tff(fact_2801_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).
% nat_less_add_iff2
tff(fact_2802_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_2803_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_2804_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).
% diff_mult_distrib2
tff(fact_2805_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).
% diff_mult_distrib
tff(fact_2806_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_2807_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
tff(fact_2808_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_2809_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% nat_mult_less_cancel1
tff(fact_2810_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
<=> ( M = N ) ) ) ).
% nat_mult_eq_cancel1
tff(fact_2811_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).
% nat_diff_add_eq2
tff(fact_2812_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).
% nat_diff_add_eq1
tff(fact_2813_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).
% nat_le_add_iff2
tff(fact_2814_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).
% nat_le_add_iff1
tff(fact_2815_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
<=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).
% nat_eq_add_iff2
tff(fact_2816_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).
% nat_eq_add_iff1
tff(fact_2817_mult__less__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ).
% mult_less_iff1
tff(fact_2818_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).
% mult_less_iff1
tff(fact_2819_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).
% mult_less_iff1
tff(fact_2820_sum__squares__eq__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = zero_z3403309356797280102nteger )
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_2821_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_2822_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_2823_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% nat_mult_le_cancel1
tff(fact_2824_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).
% nat_less_add_iff1
tff(fact_2825_nat0__intermed__int__val,axiom,
! [N: nat,F: fun_nat_int,K: int] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
=> ? [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
& ( aa_nat_int(F,I2) = K ) ) ) ) ) ).
% nat0_intermed_int_val
tff(fact_2826_slice__len,axiom,
! [From: nat,To: nat,Xs: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs)))
=> ( aa_list_nat_nat(size_size_list_nat,slice_nat(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).
% slice_len
tff(fact_2827_slice__len,axiom,
! [From: nat,To: nat,Xs: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_int_nat(size_size_list_int,Xs)))
=> ( aa_list_int_nat(size_size_list_int,slice_int(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).
% slice_len
tff(fact_2828_slice__len,axiom,
! [From: nat,To: nat,Xs: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_bool_nat(size_size_list_bool,Xs)))
=> ( aa_list_bool_nat(size_size_list_bool,slice_bool(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).
% slice_len
tff(fact_2829_ceiling__divide__lower,axiom,
! [Q2: rat,P3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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(P3),Q2)))),one_one_rat)),Q2)),P3)) ) ).
% ceiling_divide_lower
tff(fact_2830_floor__divide__upper,axiom,
! [Q2: rat,P3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P3),Q2)))),one_one_rat)),Q2))) ) ).
% floor_divide_upper
tff(fact_2831_nat__ivt__aux,axiom,
! [N: nat,F: fun_nat_int,K: int] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
=> ? [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
& ( aa_nat_int(F,I2) = K ) ) ) ) ) ).
% nat_ivt_aux
tff(fact_2832_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( member_rat(A,ring_1_Ints_rat)
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_Code_integer(ring_18347121197199848620nteger,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_Ints
tff(fact_2833_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( member_rat(A,ring_1_Ints_rat)
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_rat(ring_1_of_int_rat,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_Ints
tff(fact_2834_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( member_rat(A,ring_1_Ints_rat)
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_int(ring_1_of_int_int,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_Ints
tff(fact_2835_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( member_rat(A,ring_1_Ints_rat)
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))))) ) ) ).
% le_mult_floor_Ints
tff(fact_2836_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( member_rat(A,ring_1_Ints_rat)
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))))) ) ) ).
% le_mult_floor_Ints
tff(fact_2837_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( member_rat(A,ring_1_Ints_rat)
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_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_int_int(ring_1_of_int_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))))) ) ) ).
% le_mult_floor_Ints
tff(fact_2838_frac__unique__iff,axiom,
! [X: rat,A: rat] :
( ( aa_rat_rat(archimedean_frac_rat,X) = A )
<=> ( member_rat(aa_rat_rat(minus_minus_rat(X),A),ring_1_Ints_rat)
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) ) ) ).
% frac_unique_iff
tff(fact_2839_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_2840_abs__idempotent,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).
% abs_idempotent
tff(fact_2841_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_2842_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
tff(fact_2843_divide__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
tff(fact_2844_divide__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
tff(fact_2845_division__ring__divide__zero,axiom,
! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).
% division_ring_divide_zero
tff(fact_2846_div__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).
% div_0
tff(fact_2847_div__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).
% div_0
tff(fact_2848_div__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).
% div_0
tff(fact_2849_div__0,axiom,
! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).
% div_0
tff(fact_2850_div__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).
% div_0
tff(fact_2851_div__by__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).
% div_by_0
tff(fact_2852_div__by__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).
% div_by_0
tff(fact_2853_div__by__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% div_by_0
tff(fact_2854_div__by__0,axiom,
! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).
% div_by_0
tff(fact_2855_div__by__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).
% div_by_0
tff(fact_2856_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_2857_div__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).
% div_by_1
tff(fact_2858_div__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),one_one_Code_integer) = A ).
% div_by_1
tff(fact_2859_div__by__1,axiom,
! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).
% div_by_1
tff(fact_2860_div__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),one_one_Code_natural) = A ).
% div_by_1
tff(fact_2861_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_2862_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
tff(fact_2863_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_2864_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_2865_abs__eq__0,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
tff(fact_2866_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_2867_abs__zero,axiom,
aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
% abs_zero
tff(fact_2868_abs__zero,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% abs_zero
tff(fact_2869_abs__zero,axiom,
aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).
% abs_zero
tff(fact_2870_abs__0,axiom,
aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
% abs_0
tff(fact_2871_abs__0,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% abs_0
tff(fact_2872_abs__0,axiom,
aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).
% abs_0
tff(fact_2873_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).
% abs_add_abs
tff(fact_2874_abs__add__abs,axiom,
! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).
% abs_add_abs
tff(fact_2875_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_2876_abs__1,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_1
tff(fact_2877_abs__1,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).
% abs_1
tff(fact_2878_abs__1,axiom,
aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).
% abs_1
tff(fact_2879_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).
% nat_mult_div_cancel_disj
tff(fact_2880_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_2881_abs__minus__cancel,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).
% abs_minus_cancel
tff(fact_2882_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_2883_abs__of__nat,axiom,
! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% abs_of_nat
tff(fact_2884_abs__of__nat,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).
% abs_of_nat
tff(fact_2885_abs__of__nat,axiom,
! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).
% abs_of_nat
tff(fact_2886_slice__eq__bounds__empty,axiom,
! [I: nat,Xs: list_int] : slice_int(I,I,Xs) = nil_int ).
% slice_eq_bounds_empty
tff(fact_2887_slice__eq__bounds__empty,axiom,
! [I: nat,Xs: list_nat] : slice_nat(I,I,Xs) = nil_nat ).
% slice_eq_bounds_empty
tff(fact_2888_slice__eq__bounds__empty,axiom,
! [I: nat,Xs: list_char] : slice_char(I,I,Xs) = nil_char ).
% slice_eq_bounds_empty
tff(fact_2889_slice__Nil,axiom,
! [Begin: nat,End: nat] : slice_int(Begin,End,nil_int) = nil_int ).
% slice_Nil
tff(fact_2890_slice__Nil,axiom,
! [Begin: nat,End: nat] : slice_nat(Begin,End,nil_nat) = nil_nat ).
% slice_Nil
tff(fact_2891_slice__Nil,axiom,
! [Begin: nat,End: nat] : slice_char(Begin,End,nil_char) = nil_char ).
% slice_Nil
tff(fact_2892_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_2893_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2894_nonzero__mult__div__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2895_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2896_nonzero__mult__div__cancel__left,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2897_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_2898_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2899_nonzero__mult__div__cancel__right,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2900_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2901_nonzero__mult__div__cancel__right,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2902_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(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_2903_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(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2904_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(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2905_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(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2906_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_2907_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_2908_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_2909_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_2910_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_2911_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),A) = zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ) ).
% divide_self_if
tff(fact_2912_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).
% divide_self
tff(fact_2913_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_2914_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_2915_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_2916_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).
% div_self
tff(fact_2917_div__self,axiom,
! [A: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),A) = one_one_Code_integer ) ) ).
% div_self
tff(fact_2918_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).
% div_self
tff(fact_2919_div__self,axiom,
! [A: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),A) = one_one_Code_natural ) ) ).
% div_self
tff(fact_2920_abs__le__zero__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger))
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
tff(fact_2921_abs__le__zero__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
<=> ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
tff(fact_2922_abs__le__zero__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
<=> ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
tff(fact_2923_abs__le__self__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% abs_le_self_iff
tff(fact_2924_abs__le__self__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% abs_le_self_iff
tff(fact_2925_abs__le__self__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% abs_le_self_iff
tff(fact_2926_abs__of__nonneg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).
% abs_of_nonneg
tff(fact_2927_abs__of__nonneg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).
% abs_of_nonneg
tff(fact_2928_abs__of__nonneg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_nonneg
tff(fact_2929_zero__less__abs__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)))
<=> ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
tff(fact_2930_zero__less__abs__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
<=> ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
tff(fact_2931_zero__less__abs__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
<=> ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
tff(fact_2932_divide__minus1,axiom,
! [X: rat] : aa_rat_rat(divide_divide_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).
% divide_minus1
tff(fact_2933_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_2934_abs__neg__one,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).
% abs_neg_one
tff(fact_2935_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_2936_abs__power__int__minus,axiom,
! [A: rat,N: int] : aa_rat_rat(abs_abs_rat,aa_int_rat(power_int_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_rat_rat(abs_abs_rat,aa_int_rat(power_int_rat(A),N)) ).
% abs_power_int_minus
tff(fact_2937_of__int__abs,axiom,
! [X: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,X)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).
% of_int_abs
tff(fact_2938_of__int__abs,axiom,
! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,X)) ).
% of_int_abs
tff(fact_2939_of__int__abs,axiom,
! [X: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X)) ).
% of_int_abs
tff(fact_2940_frac__eq__0__iff,axiom,
! [X: rat] :
( ( aa_rat_rat(archimedean_frac_rat,X) = zero_zero_rat )
<=> member_rat(X,ring_1_Ints_rat) ) ).
% frac_eq_0_iff
tff(fact_2941_slice__complete,axiom,
! [Xs: list_nat] : slice_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs),Xs) = Xs ).
% slice_complete
tff(fact_2942_slice__complete,axiom,
! [Xs: list_int] : slice_int(zero_zero_nat,aa_list_int_nat(size_size_list_int,Xs),Xs) = Xs ).
% slice_complete
tff(fact_2943_slice__complete,axiom,
! [Xs: list_bool] : slice_bool(zero_zero_nat,aa_list_bool_nat(size_size_list_bool,Xs),Xs) = Xs ).
% slice_complete
tff(fact_2944_zero__le__divide__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% zero_le_divide_1_iff
tff(fact_2945_divide__le__0__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% divide_le_0_1_iff
tff(fact_2946_zero__less__divide__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% zero_less_divide_1_iff
tff(fact_2947_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% less_divide_eq_1_pos
tff(fact_2948_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).
% less_divide_eq_1_neg
tff(fact_2949_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).
% divide_less_eq_1_pos
tff(fact_2950_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% divide_less_eq_1_neg
tff(fact_2951_divide__less__0__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% divide_less_0_1_iff
tff(fact_2952_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2953_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2954_abs__of__nonpos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_nonpos
tff(fact_2955_abs__of__nonpos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_2956_abs__of__nonpos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).
% abs_of_nonpos
tff(fact_2957_zero__le__divide__abs__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
| ( B = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
tff(fact_2958_divide__le__0__abs__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
| ( B = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
tff(fact_2959_zabs__less__one__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
<=> ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
tff(fact_2960_floor__add2,axiom,
! [X: rat,Y: rat] :
( ( member_rat(X,ring_1_Ints_rat)
| member_rat(Y,ring_1_Ints_rat) )
=> ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ) ).
% floor_add2
tff(fact_2961_frac__gt__0__iff,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)))
<=> ~ member_rat(X,ring_1_Ints_rat) ) ).
% frac_gt_0_iff
tff(fact_2962_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% le_divide_eq_1_pos
tff(fact_2963_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% le_divide_eq_1_neg
tff(fact_2964_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% divide_le_eq_1_pos
tff(fact_2965_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% divide_le_eq_1_neg
tff(fact_2966_div__abs__eq__div__nat,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).
% div_abs_eq_div_nat
tff(fact_2967_zdiv__int,axiom,
! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% zdiv_int
tff(fact_2968_Ints__abs,axiom,
! [A: int] :
( member_int(A,ring_1_Ints_int)
=> member_int(aa_int_int(abs_abs_int,A),ring_1_Ints_int) ) ).
% Ints_abs
tff(fact_2969_Ints__abs,axiom,
! [A: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> member_Code_integer(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A),ring_11222124179247155820nteger) ) ).
% Ints_abs
tff(fact_2970_Ints__abs,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> member_rat(aa_rat_rat(abs_abs_rat,A),ring_1_Ints_rat) ) ).
% Ints_abs
tff(fact_2971_nonzero__abs__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).
% nonzero_abs_divide
tff(fact_2972_abs__div__pos,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).
% abs_div_pos
tff(fact_2973_abs__ge__self,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))) ).
% abs_ge_self
tff(fact_2974_abs__ge__self,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).
% abs_ge_self
tff(fact_2975_abs__ge__self,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).
% abs_ge_self
tff(fact_2976_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% abs_le_D1
tff(fact_2977_abs__le__D1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% abs_le_D1
tff(fact_2978_abs__le__D1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% abs_le_D1
tff(fact_2979_abs__eq__0__iff,axiom,
! [A: int] :
( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
tff(fact_2980_abs__eq__0__iff,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0_iff
tff(fact_2981_abs__eq__0__iff,axiom,
! [A: rat] :
( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% abs_eq_0_iff
tff(fact_2982_floor__divide__of__int__eq,axiom,
! [K: int,L: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).
% floor_divide_of_int_eq
tff(fact_2983_abs__one,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_one
tff(fact_2984_abs__one,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).
% abs_one
tff(fact_2985_abs__one,axiom,
aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).
% abs_one
tff(fact_2986_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)) ).
% abs_minus_commute
tff(fact_2987_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_2988_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_2989_add__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_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_2990_power__int__abs,axiom,
! [A: rat,N: int] : aa_rat_rat(abs_abs_rat,aa_int_rat(power_int_rat(A),N)) = aa_int_rat(power_int_rat(aa_rat_rat(abs_abs_rat,A)),N) ).
% power_int_abs
tff(fact_2991_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(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_div_as_int
tff(fact_2992_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_2993_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_2994_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
=> ( 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_NO_MATCH
tff(fact_2995_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: int,A: int,B: int] :
( pp(aa_int_bool(aa_nat_fun_int_bool(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
=> ( 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_NO_MATCH
tff(fact_2996_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_2997_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_2998_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C))
=> ( 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_NO_MATCH
tff(fact_2999_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C))
=> ( 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_NO_MATCH
tff(fact_3000_distrib__right__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,X),Y)),C))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3001_distrib__right__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,X),Y)),C))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3002_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3003_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3004_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
=> ( 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_NO_MATCH
tff(fact_3005_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: int,B: int,C: int] :
( pp(aa_int_bool(aa_nat_fun_int_bool(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
=> ( 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_NO_MATCH
tff(fact_3006_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3007_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3008_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A))
=> ( 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_NO_MATCH
tff(fact_3009_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A))
=> ( 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_NO_MATCH
tff(fact_3010_distrib__left__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,X),Y)),A))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3011_distrib__left__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(nO_MAT7795273704451493282er_rat(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,X),Y)),A))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3012_power__int__divide__distrib,axiom,
! [X: rat,Y: rat,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(divide_divide_rat(X),Y)),M) = aa_rat_rat(divide_divide_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(Y),M)) ).
% power_int_divide_distrib
tff(fact_3013_Ints__0,axiom,
member_int(zero_zero_int,ring_1_Ints_int) ).
% Ints_0
tff(fact_3014_Ints__0,axiom,
member_Code_integer(zero_z3403309356797280102nteger,ring_11222124179247155820nteger) ).
% Ints_0
tff(fact_3015_Ints__0,axiom,
member_rat(zero_zero_rat,ring_1_Ints_rat) ).
% Ints_0
tff(fact_3016_Ints__add,axiom,
! [A: code_integer,B: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> ( member_Code_integer(B,ring_11222124179247155820nteger)
=> member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),ring_11222124179247155820nteger) ) ) ).
% Ints_add
tff(fact_3017_Ints__add,axiom,
! [A: rat,B: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( member_rat(B,ring_1_Ints_rat)
=> member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),ring_1_Ints_rat) ) ) ).
% Ints_add
tff(fact_3018_Ints__add,axiom,
! [A: int,B: int] :
( member_int(A,ring_1_Ints_int)
=> ( member_int(B,ring_1_Ints_int)
=> member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B),ring_1_Ints_int) ) ) ).
% Ints_add
tff(fact_3019_Ints__mult,axiom,
! [A: code_integer,B: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> ( member_Code_integer(B,ring_11222124179247155820nteger)
=> member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),ring_11222124179247155820nteger) ) ) ).
% Ints_mult
tff(fact_3020_Ints__mult,axiom,
! [A: rat,B: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( member_rat(B,ring_1_Ints_rat)
=> member_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),ring_1_Ints_rat) ) ) ).
% Ints_mult
tff(fact_3021_Ints__mult,axiom,
! [A: int,B: int] :
( member_int(A,ring_1_Ints_int)
=> ( member_int(B,ring_1_Ints_int)
=> member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B),ring_1_Ints_int) ) ) ).
% Ints_mult
tff(fact_3022_Ints__1,axiom,
member_int(one_one_int,ring_1_Ints_int) ).
% Ints_1
tff(fact_3023_Ints__1,axiom,
member_Code_integer(one_one_Code_integer,ring_11222124179247155820nteger) ).
% Ints_1
tff(fact_3024_Ints__1,axiom,
member_rat(one_one_rat,ring_1_Ints_rat) ).
% Ints_1
tff(fact_3025_Ints__diff,axiom,
! [A: code_integer,B: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> ( member_Code_integer(B,ring_11222124179247155820nteger)
=> member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B),ring_11222124179247155820nteger) ) ) ).
% Ints_diff
tff(fact_3026_Ints__diff,axiom,
! [A: rat,B: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( member_rat(B,ring_1_Ints_rat)
=> member_rat(aa_rat_rat(minus_minus_rat(A),B),ring_1_Ints_rat) ) ) ).
% Ints_diff
tff(fact_3027_Ints__diff,axiom,
! [A: int,B: int] :
( member_int(A,ring_1_Ints_int)
=> ( member_int(B,ring_1_Ints_int)
=> member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B),ring_1_Ints_int) ) ) ).
% Ints_diff
tff(fact_3028_minus__in__Ints__iff,axiom,
! [X: int] :
( member_int(aa_int_int(uminus_uminus_int,X),ring_1_Ints_int)
<=> member_int(X,ring_1_Ints_int) ) ).
% minus_in_Ints_iff
tff(fact_3029_minus__in__Ints__iff,axiom,
! [X: code_integer] :
( member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),ring_11222124179247155820nteger)
<=> member_Code_integer(X,ring_11222124179247155820nteger) ) ).
% minus_in_Ints_iff
tff(fact_3030_minus__in__Ints__iff,axiom,
! [X: rat] :
( member_rat(aa_rat_rat(uminus_uminus_rat,X),ring_1_Ints_rat)
<=> member_rat(X,ring_1_Ints_rat) ) ).
% minus_in_Ints_iff
tff(fact_3031_Ints__minus,axiom,
! [A: int] :
( member_int(A,ring_1_Ints_int)
=> member_int(aa_int_int(uminus_uminus_int,A),ring_1_Ints_int) ) ).
% Ints_minus
tff(fact_3032_Ints__minus,axiom,
! [A: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),ring_11222124179247155820nteger) ) ).
% Ints_minus
tff(fact_3033_Ints__minus,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> member_rat(aa_rat_rat(uminus_uminus_rat,A),ring_1_Ints_rat) ) ).
% Ints_minus
tff(fact_3034_Ints__of__nat,axiom,
! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),ring_1_Ints_int) ).
% Ints_of_nat
tff(fact_3035_Ints__of__nat,axiom,
! [N: nat] : member_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N),ring_11222124179247155820nteger) ).
% Ints_of_nat
tff(fact_3036_Ints__of__nat,axiom,
! [N: nat] : member_rat(aa_nat_rat(semiri681578069525770553at_rat,N),ring_1_Ints_rat) ).
% Ints_of_nat
tff(fact_3037_Ints__of__int,axiom,
! [Z: int] : member_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Z),ring_11222124179247155820nteger) ).
% Ints_of_int
tff(fact_3038_Ints__of__int,axiom,
! [Z: int] : member_rat(aa_int_rat(ring_1_of_int_rat,Z),ring_1_Ints_rat) ).
% Ints_of_int
tff(fact_3039_Ints__of__int,axiom,
! [Z: int] : member_int(aa_int_int(ring_1_of_int_int,Z),ring_1_Ints_int) ).
% Ints_of_int
tff(fact_3040_Ints__induct,axiom,
! [Q2: code_integer,P: fun_Co8822979462182099027r_bool] :
( member_Code_integer(Q2,ring_11222124179247155820nteger)
=> ( ! [Z2: int] : pp(aa_Code_integer_bool(P,aa_int_Code_integer(ring_18347121197199848620nteger,Z2)))
=> pp(aa_Code_integer_bool(P,Q2)) ) ) ).
% Ints_induct
tff(fact_3041_Ints__induct,axiom,
! [Q2: rat,P: fun_rat_bool] :
( member_rat(Q2,ring_1_Ints_rat)
=> ( ! [Z2: int] : pp(aa_rat_bool(P,aa_int_rat(ring_1_of_int_rat,Z2)))
=> pp(aa_rat_bool(P,Q2)) ) ) ).
% Ints_induct
tff(fact_3042_Ints__induct,axiom,
! [Q2: int,P: fun_int_bool] :
( member_int(Q2,ring_1_Ints_int)
=> ( ! [Z2: int] : pp(aa_int_bool(P,aa_int_int(ring_1_of_int_int,Z2)))
=> pp(aa_int_bool(P,Q2)) ) ) ).
% Ints_induct
tff(fact_3043_Ints__cases,axiom,
! [Q2: code_integer] :
( member_Code_integer(Q2,ring_11222124179247155820nteger)
=> ~ ! [Z2: int] : Q2 != aa_int_Code_integer(ring_18347121197199848620nteger,Z2) ) ).
% Ints_cases
tff(fact_3044_Ints__cases,axiom,
! [Q2: rat] :
( member_rat(Q2,ring_1_Ints_rat)
=> ~ ! [Z2: int] : Q2 != aa_int_rat(ring_1_of_int_rat,Z2) ) ).
% Ints_cases
tff(fact_3045_Ints__cases,axiom,
! [Q2: int] :
( member_int(Q2,ring_1_Ints_int)
=> ~ ! [Z2: int] : Q2 != aa_int_int(ring_1_of_int_int,Z2) ) ).
% Ints_cases
tff(fact_3046_Ints__nonzero__abs__ge1,axiom,
! [X: code_integer] :
( member_Code_integer(X,ring_11222124179247155820nteger)
=> ( ( X != zero_z3403309356797280102nteger )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X))) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_3047_Ints__nonzero__abs__ge1,axiom,
! [X: rat] :
( member_rat(X,ring_1_Ints_rat)
=> ( ( X != zero_zero_rat )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X))) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_3048_Ints__nonzero__abs__ge1,axiom,
! [X: int] :
( member_int(X,ring_1_Ints_int)
=> ( ( X != zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X))) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_3049_Ints__nonzero__abs__less1,axiom,
! [X: code_integer] :
( member_Code_integer(X,ring_11222124179247155820nteger)
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer))
=> ( X = zero_z3403309356797280102nteger ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_3050_Ints__nonzero__abs__less1,axiom,
! [X: rat] :
( member_rat(X,ring_1_Ints_rat)
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat))
=> ( X = zero_zero_rat ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_3051_Ints__nonzero__abs__less1,axiom,
! [X: int] :
( member_int(X,ring_1_Ints_int)
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int))
=> ( X = zero_zero_int ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_3052_Ints__eq__abs__less1,axiom,
! [X: code_integer,Y: code_integer] :
( member_Code_integer(X,ring_11222124179247155820nteger)
=> ( member_Code_integer(Y,ring_11222124179247155820nteger)
=> ( ( X = Y )
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Y))),one_one_Code_integer)) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_3053_Ints__eq__abs__less1,axiom,
! [X: rat,Y: rat] :
( member_rat(X,ring_1_Ints_rat)
=> ( member_rat(Y,ring_1_Ints_rat)
=> ( ( X = Y )
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),Y))),one_one_rat)) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_3054_Ints__eq__abs__less1,axiom,
! [X: int,Y: int] :
( member_int(X,ring_1_Ints_int)
=> ( member_int(Y,ring_1_Ints_int)
=> ( ( X = Y )
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int)) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_3055_floor__divide__of__nat__eq,axiom,
! [M: nat,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).
% floor_divide_of_nat_eq
tff(fact_3056_abs__ge__zero,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))) ).
% abs_ge_zero
tff(fact_3057_abs__ge__zero,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).
% abs_ge_zero
tff(fact_3058_abs__ge__zero,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).
% abs_ge_zero
tff(fact_3059_abs__not__less__zero,axiom,
! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger)) ).
% abs_not_less_zero
tff(fact_3060_abs__not__less__zero,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).
% abs_not_less_zero
tff(fact_3061_abs__not__less__zero,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).
% abs_not_less_zero
tff(fact_3062_abs__of__pos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).
% abs_of_pos
tff(fact_3063_abs__of__pos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).
% abs_of_pos
tff(fact_3064_abs__of__pos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_pos
tff(fact_3065_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))) ).
% abs_triangle_ineq
tff(fact_3066_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).
% abs_triangle_ineq
tff(fact_3067_abs__triangle__ineq,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).
% abs_triangle_ineq
tff(fact_3068_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A)))) ).
% abs_triangle_ineq2_sym
tff(fact_3069_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3070_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)))) ).
% abs_triangle_ineq2_sym
tff(fact_3071_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))) ).
% abs_triangle_ineq3
tff(fact_3072_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3073_abs__triangle__ineq3,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).
% abs_triangle_ineq3
tff(fact_3074_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)))) ).
% abs_triangle_ineq2
tff(fact_3075_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3076_abs__triangle__ineq2,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).
% abs_triangle_ineq2
tff(fact_3077_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)) ) ) ).
% abs_leI
tff(fact_3078_abs__leI,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).
% abs_leI
tff(fact_3079_abs__leI,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).
% abs_leI
tff(fact_3080_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).
% abs_le_D2
tff(fact_3081_abs__le__D2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).
% abs_le_D2
tff(fact_3082_abs__le__D2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).
% abs_le_D2
tff(fact_3083_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).
% abs_le_iff
tff(fact_3084_abs__le__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).
% abs_le_iff
tff(fact_3085_abs__le__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).
% abs_le_iff
tff(fact_3086_abs__ge__minus__self,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))) ).
% abs_ge_minus_self
tff(fact_3087_abs__ge__minus__self,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).
% abs_ge_minus_self
tff(fact_3088_abs__ge__minus__self,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).
% abs_ge_minus_self
tff(fact_3089_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C))) ) ) ).
% divide_right_mono_neg
tff(fact_3090_divide__nonpos__nonpos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% divide_nonpos_nonpos
tff(fact_3091_divide__nonpos__nonneg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).
% divide_nonpos_nonneg
tff(fact_3092_divide__nonneg__nonpos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).
% divide_nonneg_nonpos
tff(fact_3093_divide__nonneg__nonneg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% divide_nonneg_nonneg
tff(fact_3094_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).
% zero_le_divide_iff
tff(fact_3095_divide__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).
% divide_right_mono
tff(fact_3096_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).
% divide_le_0_iff
tff(fact_3097_divide__neg__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% divide_neg_neg
tff(fact_3098_divide__neg__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).
% divide_neg_pos
tff(fact_3099_divide__pos__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).
% divide_pos_neg
tff(fact_3100_divide__pos__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% divide_pos_pos
tff(fact_3101_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).
% divide_less_0_iff
tff(fact_3102_divide__less__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
tff(fact_3103_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).
% zero_less_divide_iff
tff(fact_3104_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).
% divide_strict_right_mono
tff(fact_3105_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).
% divide_strict_right_mono_neg
tff(fact_3106_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(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_3107_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(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_3108_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
=> ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).
% eq_divide_imp
tff(fact_3109_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
=> ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).
% divide_eq_imp
tff(fact_3110_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = aa_rat_rat(divide_divide_rat(B),C) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
tff(fact_3111_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( aa_rat_rat(divide_divide_rat(B),C) = A )
<=> ( ( ( C != zero_zero_rat )
=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
tff(fact_3112_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(divide_divide_rat(W),Z) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_3113_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_3114_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% nonzero_minus_divide_right
tff(fact_3115_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).
% nonzero_minus_divide_divide
tff(fact_3116_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
=> ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).
% abs_zmult_eq_1
tff(fact_3117_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).
% div_neg_pos_less0
tff(fact_3118_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).
% neg_imp_zdiv_neg_iff
tff(fact_3119_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% pos_imp_zdiv_neg_iff
tff(fact_3120_power__int__one__over,axiom,
! [X: rat,N: int] : aa_int_rat(power_int_rat(aa_rat_rat(divide_divide_rat(one_one_rat),X)),N) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_int_rat(power_int_rat(X),N)) ).
% power_int_one_over
tff(fact_3121_Ints__double__eq__0__iff,axiom,
! [A: int] :
( member_int(A,ring_1_Ints_int)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ) ).
% Ints_double_eq_0_iff
tff(fact_3122_Ints__double__eq__0__iff,axiom,
! [A: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ) ).
% Ints_double_eq_0_iff
tff(fact_3123_Ints__double__eq__0__iff,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ) ).
% Ints_double_eq_0_iff
tff(fact_3124_dense__eq0__I,axiom,
! [X: rat] :
( ! [E3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E3)) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
tff(fact_3125_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).
% abs_eq_mult
tff(fact_3126_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
=> ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).
% abs_eq_mult
tff(fact_3127_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
=> ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).
% abs_eq_mult
tff(fact_3128_abs__mult__pos,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),X) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_3129_abs__mult__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_3130_abs__mult__pos,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_3131_abs__minus__le__zero,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))),zero_z3403309356797280102nteger)) ).
% abs_minus_le_zero
tff(fact_3132_abs__minus__le__zero,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3133_abs__minus__le__zero,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).
% abs_minus_le_zero
tff(fact_3134_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B) )
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
& ( ( B = A )
| ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_3135_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(abs_abs_rat,B) )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& ( ( B = A )
| ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_3136_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(abs_abs_int,B) )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& ( ( B = A )
| ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_3137_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = B )
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
& ( ( A = B )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_3138_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(abs_abs_rat,A) = B )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
& ( ( A = B )
| ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_3139_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( aa_int_int(abs_abs_int,A) = B )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
& ( ( A = B )
| ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_3140_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D))))) ).
% abs_diff_triangle_ineq
tff(fact_3141_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_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_3142_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))))) ).
% abs_diff_triangle_ineq
tff(fact_3143_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))) ).
% abs_triangle_ineq4
tff(fact_3144_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).
% abs_triangle_ineq4
tff(fact_3145_abs__triangle__ineq4,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).
% abs_triangle_ineq4
tff(fact_3146_abs__diff__le__iff,axiom,
! [X: code_integer,A: code_integer,R: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),A))),R))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R)),X))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R))) ) ) ).
% abs_diff_le_iff
tff(fact_3147_abs__diff__le__iff,axiom,
! [X: rat,A: rat,R: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R))) ) ) ).
% abs_diff_le_iff
tff(fact_3148_abs__diff__le__iff,axiom,
! [X: int,A: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),X))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R))) ) ) ).
% abs_diff_le_iff
tff(fact_3149_abs__if,axiom,
! [A: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = A ) ) ) ).
% abs_if
tff(fact_3150_abs__if,axiom,
! [A: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ) ).
% abs_if
tff(fact_3151_abs__if,axiom,
! [A: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).
% abs_if
tff(fact_3152_abs__of__neg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).
% abs_of_neg
tff(fact_3153_abs__of__neg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_neg
tff(fact_3154_abs__of__neg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3155_abs__if__raw,axiom,
! [X3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),zero_zero_int))
=> ( aa_int_int(abs_abs_int,X3) = aa_int_int(uminus_uminus_int,X3) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),zero_zero_int))
=> ( aa_int_int(abs_abs_int,X3) = X3 ) ) ) ).
% abs_if_raw
tff(fact_3156_abs__if__raw,axiom,
! [X3: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3) = X3 ) ) ) ).
% abs_if_raw
tff(fact_3157_abs__if__raw,axiom,
! [X3: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),zero_zero_rat))
=> ( aa_rat_rat(abs_abs_rat,X3) = aa_rat_rat(uminus_uminus_rat,X3) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),zero_zero_rat))
=> ( aa_rat_rat(abs_abs_rat,X3) = X3 ) ) ) ).
% abs_if_raw
tff(fact_3158_abs__diff__less__iff,axiom,
! [X: code_integer,A: code_integer,R: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),A))),R))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R)),X))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R))) ) ) ).
% abs_diff_less_iff
tff(fact_3159_abs__diff__less__iff,axiom,
! [X: rat,A: rat,R: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R))) ) ) ).
% abs_diff_less_iff
tff(fact_3160_abs__diff__less__iff,axiom,
! [X: int,A: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),X))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R))) ) ) ).
% abs_diff_less_iff
tff(fact_3161_div__positive,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))) ) ) ).
% div_positive
tff(fact_3162_div__positive,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) ) ) ).
% div_positive
tff(fact_3163_div__positive,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).
% div_positive
tff(fact_3164_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = zero_z3403309356797280102nteger ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3165_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3166_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3167_divide__nonpos__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).
% divide_nonpos_pos
tff(fact_3168_divide__nonpos__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% divide_nonpos_neg
tff(fact_3169_divide__nonneg__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% divide_nonneg_pos
tff(fact_3170_divide__nonneg__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).
% divide_nonneg_neg
tff(fact_3171_divide__le__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% divide_le_cancel
tff(fact_3172_frac__less2,axiom,
! [X: rat,Y: rat,W: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).
% frac_less2
tff(fact_3173_frac__less,axiom,
! [X: rat,Y: rat,W: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).
% frac_less
tff(fact_3174_frac__le,axiom,
! [Y: rat,X: rat,W: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).
% frac_le
tff(fact_3175_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3176_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3177_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3178_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% divide_less_eq
tff(fact_3179_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% less_divide_eq
tff(fact_3180_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).
% neg_divide_less_eq
tff(fact_3181_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% neg_less_divide_eq
tff(fact_3182_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% pos_divide_less_eq
tff(fact_3183_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).
% pos_less_divide_eq
tff(fact_3184_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).
% mult_imp_div_pos_less
tff(fact_3185_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% mult_imp_less_div_pos
tff(fact_3186_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3187_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3188_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% less_divide_eq_1
tff(fact_3189_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
tff(fact_3190_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Z)),B) = B ) )
& ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(2)
tff(fact_3191_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
& ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z) ) ) ) ).
% add_divide_eq_if_simps(1)
tff(fact_3192_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_3193_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).
% add_frac_num
tff(fact_3194_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).
% add_num_frac
tff(fact_3195_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).
% add_divide_eq_iff
tff(fact_3196_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).
% divide_add_eq_iff
tff(fact_3197_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_3198_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_3199_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_3200_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
& ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z) ) ) ) ).
% add_divide_eq_if_simps(4)
tff(fact_3201_less__half__sum,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)))) ) ).
% less_half_sum
tff(fact_3202_gt__half__sum,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B)) ) ).
% gt_half_sum
tff(fact_3203_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)) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_3204_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 )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_3205_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(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_3206_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(aa_rat_fun_rat_rat(times_times_rat,C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_3207_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_3208_zabs__def,axiom,
! [I: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
=> ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
=> ( aa_int_int(abs_abs_int,I) = I ) ) ) ).
% zabs_def
tff(fact_3209_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).
% nat_mult_div_cancel1
tff(fact_3210_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_3211_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
tff(fact_3212_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
% pos_imp_zdiv_nonneg_iff
tff(fact_3213_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).
% neg_imp_zdiv_nonneg_iff
tff(fact_3214_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).
% pos_imp_zdiv_pos_iff
tff(fact_3215_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).
% div_nonpos_pos_le0
tff(fact_3216_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).
% div_nonneg_neg_le0
tff(fact_3217_div__positive__int,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).
% div_positive_int
tff(fact_3218_div__int__pos__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)))
<=> ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).
% div_int_pos_iff
tff(fact_3219_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B4)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).
% zdiv_mono2_neg
tff(fact_3220_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).
% zdiv_mono1_neg
tff(fact_3221_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
<=> ( ( K = zero_zero_int )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).
% zdiv_eq_0_iff
tff(fact_3222_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B4))) ) ) ) ).
% zdiv_mono2
tff(fact_3223_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B))) ) ) ).
% zdiv_mono1
tff(fact_3224_int__div__less__self,axiom,
! [X: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X)) ) ) ).
% int_div_less_self
tff(fact_3225_power__int__diff,axiom,
! [X: rat,M: int,N: int] :
( ( ( X != zero_zero_rat )
| ( M != N ) )
=> ( aa_int_rat(power_int_rat(X),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)) = aa_rat_rat(divide_divide_rat(aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(X),N)) ) ) ).
% power_int_diff
tff(fact_3226_Ints__odd__nonzero,axiom,
! [A: int] :
( member_int(A,ring_1_Ints_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),A)),A) != zero_zero_int ) ) ).
% Ints_odd_nonzero
tff(fact_3227_Ints__odd__nonzero,axiom,
! [A: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),A) != zero_z3403309356797280102nteger ) ) ).
% Ints_odd_nonzero
tff(fact_3228_Ints__odd__nonzero,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).
% Ints_odd_nonzero
tff(fact_3229_abs__add__one__gt__zero,axiom,
! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)))) ).
% abs_add_one_gt_zero
tff(fact_3230_abs__add__one__gt__zero,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).
% abs_add_one_gt_zero
tff(fact_3231_abs__add__one__gt__zero,axiom,
! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).
% abs_add_one_gt_zero
tff(fact_3232_of__int__leD,axiom,
! [N: int,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).
% of_int_leD
tff(fact_3233_of__int__leD,axiom,
! [N: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).
% of_int_leD
tff(fact_3234_of__int__leD,axiom,
! [N: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).
% of_int_leD
tff(fact_3235_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).
% of_int_lessD
tff(fact_3236_of__int__lessD,axiom,
! [N: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).
% of_int_lessD
tff(fact_3237_of__int__lessD,axiom,
! [N: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).
% of_int_lessD
tff(fact_3238_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% divide_le_eq
tff(fact_3239_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% le_divide_eq
tff(fact_3240_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3241_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).
% neg_divide_le_eq
tff(fact_3242_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% neg_le_divide_eq
tff(fact_3243_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% pos_divide_le_eq
tff(fact_3244_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B)) ) ) ).
% pos_le_divide_eq
tff(fact_3245_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).
% mult_imp_div_pos_le
tff(fact_3246_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).
% mult_imp_le_div_pos
tff(fact_3247_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3248_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% le_divide_eq_1
tff(fact_3249_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
tff(fact_3250_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat)) ) ) ) ).
% frac_le_eq
tff(fact_3251_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat)) ) ) ) ).
% frac_less_eq
tff(fact_3252_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% pos_minus_divide_less_eq
tff(fact_3253_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).
% pos_less_minus_divide_eq
tff(fact_3254_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).
% neg_minus_divide_less_eq
tff(fact_3255_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% neg_less_minus_divide_eq
tff(fact_3256_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% minus_divide_less_eq
tff(fact_3257_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% less_minus_divide_eq
tff(fact_3258_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_3259_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = B ) )
& ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(3)
tff(fact_3260_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
& ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(6)
tff(fact_3261_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
& ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(5)
tff(fact_3262_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: 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(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_3263_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).
% nat_abs_triangle_ineq
tff(fact_3264_div__geq,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).
% div_geq
tff(fact_3265_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N))) ) ) ).
% verit_less_mono_div_int2
tff(fact_3266_div__eq__minus1,axiom,
! [B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).
% div_eq_minus1
tff(fact_3267_Ints__odd__less__0,axiom,
! [A: code_integer] :
( member_Code_integer(A,ring_11222124179247155820nteger)
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ) ).
% Ints_odd_less_0
tff(fact_3268_Ints__odd__less__0,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).
% Ints_odd_less_0
tff(fact_3269_Ints__odd__less__0,axiom,
! [A: int] :
( member_int(A,ring_1_Ints_int)
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% Ints_odd_less_0
tff(fact_3270_scaling__mono,axiom,
! [U: rat,V: rat,R: rat,S: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_rat_rat(minus_minus_rat(V),U))),S))),V)) ) ) ) ).
% scaling_mono
tff(fact_3271_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% pos_minus_divide_le_eq
tff(fact_3272_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).
% pos_le_minus_divide_eq
tff(fact_3273_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).
% neg_minus_divide_le_eq
tff(fact_3274_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) ) ) ).
% neg_le_minus_divide_eq
tff(fact_3275_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% minus_divide_le_eq
tff(fact_3276_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% le_minus_divide_eq
tff(fact_3277_nat__approx__posE,axiom,
! [E: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
=> ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E)) ) ).
% nat_approx_posE
tff(fact_3278_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).
% nat_abs_int_diff
tff(fact_3279_split__zdiv,axiom,
! [P: fun_int_bool,N: int,K: int] :
( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
<=> ( ( ( K = zero_zero_int )
=> pp(aa_int_bool(P,zero_zero_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ! [I3: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
=> pp(aa_int_bool(P,I3)) ) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ! [I3: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
=> pp(aa_int_bool(P,I3)) ) ) ) ) ).
% split_zdiv
tff(fact_3280_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).
% int_div_neg_eq
tff(fact_3281_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).
% int_div_pos_eq
tff(fact_3282_frac__neg,axiom,
! [X: rat] :
( ( member_rat(X,ring_1_Ints_rat)
=> ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = zero_zero_rat ) )
& ( ~ member_rat(X,ring_1_Ints_rat)
=> ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,X)) ) ) ) ).
% frac_neg
tff(fact_3283_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: fun_nat_int,K: int] :
( ! [I2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
=> ? [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
& ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).
% nat_intermed_int_val
tff(fact_3284_floor__divide__lower,axiom,
! [Q2: rat,P3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P3),Q2)))),Q2)),P3)) ) ).
% floor_divide_lower
tff(fact_3285_ceiling__divide__upper,axiom,
! [Q2: rat,P3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q2)))),Q2))) ) ).
% ceiling_divide_upper
tff(fact_3286_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).
% decr_lemma
tff(fact_3287_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).
% incr_lemma
tff(fact_3288_div__pos__geq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).
% div_pos_geq
tff(fact_3289_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_3290_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self1
tff(fact_3291_div__mult__self1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).
% div_mult_self1
tff(fact_3292_div__mult__self1,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)) ) ) ).
% div_mult_self1
tff(fact_3293_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_3294_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self2
tff(fact_3295_div__mult__self2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).
% div_mult_self2
tff(fact_3296_div__mult__self2,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)) ) ) ).
% div_mult_self2
tff(fact_3297_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_3298_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self3
tff(fact_3299_div__mult__self3,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).
% div_mult_self3
tff(fact_3300_div__mult__self3,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)),A)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)) ) ) ).
% div_mult_self3
tff(fact_3301_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_3302_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self4
tff(fact_3303_div__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).
% div_mult_self4
tff(fact_3304_div__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),A)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)) ) ) ).
% div_mult_self4
tff(fact_3305_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).
% div_pos_pos_trivial
tff(fact_3306_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).
% div_neg_neg_trivial
tff(fact_3307_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).
% div_mult_mult1_if
tff(fact_3308_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).
% div_mult_mult1_if
tff(fact_3309_div__mult__mult1__if,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ( C = zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = zero_z3403309356797280102nteger ) )
& ( ( C != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ) ).
% div_mult_mult1_if
tff(fact_3310_div__mult__mult1__if,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( ( C = zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) = zero_z2226904508553997617atural ) )
& ( ( C != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B) ) ) ) ).
% div_mult_mult1_if
tff(fact_3311_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_3312_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).
% div_mult_mult2
tff(fact_3313_div__mult__mult2,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ).
% div_mult_mult2
tff(fact_3314_div__mult__mult2,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B) ) ) ).
% div_mult_mult2
tff(fact_3315_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_3316_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).
% div_mult_mult1
tff(fact_3317_div__mult__mult1,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) ) ) ).
% div_mult_mult1
tff(fact_3318_div__mult__mult1,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B) ) ) ).
% div_mult_mult1
tff(fact_3319_div__minus1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).
% div_minus1_right
tff(fact_3320_div__minus1__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% div_minus1_right
tff(fact_3321_div__by__Suc__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).
% div_by_Suc_0
tff(fact_3322_div__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).
% div_less
tff(fact_3323_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).
% div_mult_self_is_m
tff(fact_3324_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).
% div_mult_self1_is_m
tff(fact_3325_div__le__dividend,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).
% div_le_dividend
tff(fact_3326_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).
% div_le_mono
tff(fact_3327_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
tff(fact_3328_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).
% Suc_div_le_mono
tff(fact_3329_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3330_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3331_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3332_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3333_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).
% div_times_less_eq_dividend
tff(fact_3334_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).
% times_div_less_eq_dividend
tff(fact_3335_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% div_mult2_eq'
tff(fact_3336_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% div_mult2_eq'
tff(fact_3337_div__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% div_mult2_eq'
tff(fact_3338_div__mult2__eq_H,axiom,
! [A: code_natural,M: nat,N: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),aa_nat_Code_natural(semiri3763490453095760265atural,M))),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% div_mult2_eq'
tff(fact_3339_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).
% div_le_mono2
tff(fact_3340_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% div_greater_zero_iff
tff(fact_3341_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2))) ) ) ).
% div_less_iff_less_mult
tff(fact_3342_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
<=> ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
tff(fact_3343_div__less__dividend,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).
% div_less_dividend
tff(fact_3344_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% zdiv_zmult2_eq
tff(fact_3345_nat__div__distrib,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).
% nat_div_distrib
tff(fact_3346_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).
% nat_div_distrib'
tff(fact_3347_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_3348_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% div_add_self2
tff(fact_3349_div__add__self2,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),one_one_Code_integer) ) ) ).
% div_add_self2
tff(fact_3350_div__add__self2,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)),one_one_Code_natural) ) ) ).
% div_add_self2
tff(fact_3351_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_3352_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% div_add_self1
tff(fact_3353_div__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),one_one_Code_integer) ) ) ).
% div_add_self1
tff(fact_3354_div__add__self1,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)),one_one_Code_natural) ) ) ).
% div_add_self1
tff(fact_3355_div__if,axiom,
! [M: nat,N: nat] :
( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
& ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).
% div_if
tff(fact_3356_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),N)) ) ) ).
% less_eq_div_iff_mult_less_eq
tff(fact_3357_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q2))))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q2 ) ) ) ).
% div_nat_eqI
tff(fact_3358_split__div,axiom,
! [P: fun_nat_bool,M: nat,N: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
<=> ( ( ( N = zero_zero_nat )
=> pp(aa_nat_bool(P,zero_zero_nat)) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
=> ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J3) )
=> pp(aa_nat_bool(P,I3)) ) ) ) ) ) ).
% split_div
tff(fact_3359_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).
% dividend_less_div_times
tff(fact_3360_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).
% dividend_less_times_div
tff(fact_3361_le__div__geq,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).
% le_div_geq
tff(fact_3362_split__div_H,axiom,
! [P: fun_nat_bool,M: nat,N: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
<=> ( ( ( N = zero_zero_nat )
& pp(aa_nat_bool(P,zero_zero_nat)) )
| ? [Q6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q6)),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q6))))
& pp(aa_nat_bool(P,Q6)) ) ) ) ).
% split_div'
tff(fact_3363_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% div_pos_neg_trivial
tff(fact_3364_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_3365_bits__div__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).
% bits_div_by_1
tff(fact_3366_bits__div__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),one_one_Code_integer) = A ).
% bits_div_by_1
tff(fact_3367_bits__div__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),one_one_Code_natural) = A ).
% bits_div_by_1
tff(fact_3368_bits__div__by__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).
% bits_div_by_0
tff(fact_3369_bits__div__by__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).
% bits_div_by_0
tff(fact_3370_bits__div__by__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% bits_div_by_0
tff(fact_3371_bits__div__by__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).
% bits_div_by_0
tff(fact_3372_bits__div__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).
% bits_div_0
tff(fact_3373_bits__div__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).
% bits_div_0
tff(fact_3374_bits__div__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).
% bits_div_0
tff(fact_3375_bits__div__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).
% bits_div_0
tff(fact_3376_int__power__div__base,axiom,
! [M: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).
% int_power_div_base
tff(fact_3377_gbinomial__absorption_H,axiom,
! [K: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_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_3378_pochhammer__absorb__comp,axiom,
! [R: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R)),one_one_int)),K)) ).
% pochhammer_absorb_comp
tff(fact_3379_pochhammer__absorb__comp,axiom,
! [R: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),one_one_Code_integer)),K)) ).
% pochhammer_absorb_comp
tff(fact_3380_pochhammer__absorb__comp,axiom,
! [R: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(R),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R)),one_one_rat)),K)) ).
% pochhammer_absorb_comp
tff(fact_3381_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_3382_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_3383_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_3384_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( semiri2447717529341329178atural(N) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)),semiri2447717529341329178atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_3385_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_3386_power__one,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ).
% power_one
tff(fact_3387_power__one,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).
% power_one
tff(fact_3388_power__one,axiom,
! [N: nat] : power_7079662738309270450atural(one_one_Code_natural,N) = one_one_Code_natural ).
% power_one
tff(fact_3389_power__one,axiom,
! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).
% power_one
tff(fact_3390_power__one,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).
% power_one
tff(fact_3391_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,X) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3392_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,X) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3393_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_Code_integer(semiri4939895301339042750nteger,X) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3394_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,X) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3395_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_Code_natural(semiri3763490453095760265atural,X) = power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,B),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3396_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3397_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3398_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3399_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3400_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,B),W) = aa_nat_Code_natural(semiri3763490453095760265atural,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3401_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),N) ).
% of_nat_power
tff(fact_3402_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).
% of_nat_power
tff(fact_3403_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),N) ).
% of_nat_power
tff(fact_3404_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),N) ).
% of_nat_power
tff(fact_3405_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),N) ).
% of_nat_power
tff(fact_3406_power__one__right,axiom,
! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).
% power_one_right
tff(fact_3407_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_3408_of__nat__fact,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_int(semiri1406184849735516958ct_int,N) ).
% of_nat_fact
tff(fact_3409_of__nat__fact,axiom,
! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_Code_integer(semiri3624122377584611663nteger,N) ).
% of_nat_fact
tff(fact_3410_of__nat__fact,axiom,
! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_rat(semiri773545260158071498ct_rat,N) ).
% of_nat_fact
tff(fact_3411_of__nat__fact,axiom,
! [N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = semiri2447717529341329178atural(N) ).
% of_nat_fact
tff(fact_3412_of__nat__fact,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).
% of_nat_fact
tff(fact_3413_power__inject__exp,axiom,
! [A: code_natural,M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> ( ( power_7079662738309270450atural(A,M) = power_7079662738309270450atural(A,N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_3414_power__inject__exp,axiom,
! [A: code_integer,M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_3415_power__inject__exp,axiom,
! [A: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_3416_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_3417_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_3418_power__0__Suc,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,N)) = zero_z3403309356797280102nteger ).
% power_0_Suc
tff(fact_3419_power__0__Suc,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).
% power_0_Suc
tff(fact_3420_power__0__Suc,axiom,
! [N: nat] : power_7079662738309270450atural(zero_z2226904508553997617atural,aa_nat_nat(suc,N)) = zero_z2226904508553997617atural ).
% power_0_Suc
tff(fact_3421_power__0__Suc,axiom,
! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).
% power_0_Suc
tff(fact_3422_power__0__Suc,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).
% power_0_Suc
tff(fact_3423_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_3424_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_3425_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_3426_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_3427_gbinomial__0_I2_J,axiom,
! [K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,K)) = zero_z3403309356797280102nteger ).
% gbinomial_0(2)
tff(fact_3428_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_3429_gbinomial__0_I2_J,axiom,
! [K: nat] : gbinom7368847122466276068atural(zero_z2226904508553997617atural,aa_nat_nat(suc,K)) = zero_z2226904508553997617atural ).
% gbinomial_0(2)
tff(fact_3430_fact__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).
% fact_0
tff(fact_3431_fact__0,axiom,
aa_nat_Code_integer(semiri3624122377584611663nteger,zero_zero_nat) = one_one_Code_integer ).
% fact_0
tff(fact_3432_fact__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).
% fact_0
tff(fact_3433_fact__0,axiom,
semiri2447717529341329178atural(zero_zero_nat) = one_one_Code_natural ).
% fact_0
tff(fact_3434_fact__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).
% fact_0
tff(fact_3435_gbinomial__0_I1_J,axiom,
! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).
% gbinomial_0(1)
tff(fact_3436_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_3437_gbinomial__0_I1_J,axiom,
! [A: code_integer] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),zero_zero_nat) = one_one_Code_integer ).
% gbinomial_0(1)
tff(fact_3438_gbinomial__0_I1_J,axiom,
! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).
% gbinomial_0(1)
tff(fact_3439_gbinomial__0_I1_J,axiom,
! [A: code_natural] : gbinom7368847122466276068atural(A,zero_zero_nat) = one_one_Code_natural ).
% gbinomial_0(1)
tff(fact_3440_pochhammer__0,axiom,
! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).
% pochhammer_0
tff(fact_3441_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_3442_pochhammer__0,axiom,
! [A: code_integer] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),zero_zero_nat) = one_one_Code_integer ).
% pochhammer_0
tff(fact_3443_pochhammer__0,axiom,
! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).
% pochhammer_0
tff(fact_3444_pochhammer__0,axiom,
! [A: code_natural] : comm_s7406298101470620109atural(A,zero_zero_nat) = one_one_Code_natural ).
% pochhammer_0
tff(fact_3445_fact__1,axiom,
aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).
% fact_1
tff(fact_3446_fact__1,axiom,
aa_nat_Code_integer(semiri3624122377584611663nteger,one_one_nat) = one_one_Code_integer ).
% fact_1
tff(fact_3447_fact__1,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).
% fact_1
tff(fact_3448_fact__1,axiom,
semiri2447717529341329178atural(one_one_nat) = one_one_Code_natural ).
% fact_1
tff(fact_3449_fact__1,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).
% fact_1
tff(fact_3450_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(power_power_int(Z),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)),N) ).
% of_int_power
tff(fact_3451_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).
% of_int_power
tff(fact_3452_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).
% of_int_power
tff(fact_3453_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) = aa_int_Code_integer(ring_18347121197199848620nteger,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_3454_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_3455_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_3456_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,X) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_3457_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_3458_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_3459_power__strict__increasing__iff,axiom,
! [B: code_natural,X: nat,Y: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(B,X)),power_7079662738309270450atural(B,Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_3460_power__strict__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_3461_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_3462_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_3463_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_3464_minus__one__mult__self,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).
% minus_one_mult_self
tff(fact_3465_minus__one__mult__self,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).
% minus_one_mult_self
tff(fact_3466_minus__one__mult__self,axiom,
! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).
% minus_one_mult_self
tff(fact_3467_left__minus__one__mult__self,axiom,
! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_3468_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_3469_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_3470_power__eq__0__iff,axiom,
! [A: code_integer,N: nat] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = zero_z3403309356797280102nteger )
<=> ( ( A = zero_z3403309356797280102nteger )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_3471_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_3472_power__eq__0__iff,axiom,
! [A: code_natural,N: nat] :
( ( power_7079662738309270450atural(A,N) = zero_z2226904508553997617atural )
<=> ( ( A = zero_z2226904508553997617atural )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_3473_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
<=> ( ( A = zero_zero_int )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_3474_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_3475_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_3476_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_3477_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_3478_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_3479_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,B),W)),aa_nat_Code_natural(semiri3763490453095760265atural,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_3480_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_3481_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_3482_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_3483_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_3484_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,B),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_3485_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_3486_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,B),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_3487_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_3488_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_3489_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_3490_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_3491_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,B),W)),aa_nat_Code_natural(semiri3763490453095760265atural,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_3492_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_3493_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_3494_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_3495_fact__Suc__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).
% fact_Suc_0
tff(fact_3496_fact__Suc__0,axiom,
aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).
% fact_Suc_0
tff(fact_3497_fact__Suc__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).
% fact_Suc_0
tff(fact_3498_fact__Suc__0,axiom,
semiri2447717529341329178atural(aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_natural ).
% fact_Suc_0
tff(fact_3499_fact__Suc__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% fact_Suc_0
tff(fact_3500_fact__Suc,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% fact_Suc
tff(fact_3501_fact__Suc,axiom,
! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).
% fact_Suc
tff(fact_3502_fact__Suc,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% fact_Suc
tff(fact_3503_fact__Suc,axiom,
! [N: nat] : semiri2447717529341329178atural(aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(suc,N))),semiri2447717529341329178atural(N)) ).
% fact_Suc
tff(fact_3504_fact__Suc,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).
% fact_Suc
tff(fact_3505_power__strict__decreasing__iff,axiom,
! [B: code_natural,M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),one_one_Code_natural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(B,M)),power_7079662738309270450atural(B,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3506_power__strict__decreasing__iff,axiom,
! [B: code_integer,M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3507_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3508_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3509_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3510_power__mono__iff,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_3511_power__mono__iff,axiom,
! [A: code_natural,B: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(B,N)))
<=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_3512_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_3513_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_3514_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_3515_power__increasing__iff,axiom,
! [B: code_natural,X: nat,Y: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(B,X)),power_7079662738309270450atural(B,Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_3516_power__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_3517_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_3518_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_3519_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_3520_zero__less__power__abs__iff,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N)))
<=> ( ( A != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_3521_zero__less__power__abs__iff,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N)))
<=> ( ( A != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_3522_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
<=> ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_3523_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_3524_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_3525_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_3526_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_3527_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_3528_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_3529_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_3530_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_3531_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_3532_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_3533_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_3534_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_3535_power__decreasing__iff,axiom,
! [B: code_natural,M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,B),one_one_Code_natural))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(B,M)),power_7079662738309270450atural(B,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_3536_power__decreasing__iff,axiom,
! [B: code_integer,M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_3537_power__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_3538_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_3539_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_3540_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_3541_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_3542_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_3543_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_3544_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),power_7079662738309270450atural(aa_nat_Code_natural(semiri3763490453095760265atural,X),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_3545_pochhammer__of__nat,axiom,
! [X: nat,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_nat_int(semiri1314217659103216013at_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).
% pochhammer_of_nat
tff(fact_3546_pochhammer__of__nat,axiom,
! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).
% pochhammer_of_nat
tff(fact_3547_pochhammer__of__nat,axiom,
! [X: nat,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).
% pochhammer_of_nat
tff(fact_3548_pochhammer__of__nat,axiom,
! [X: nat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).
% pochhammer_of_nat
tff(fact_3549_pochhammer__of__nat,axiom,
! [X: nat,N: nat] : comm_s7406298101470620109atural(aa_nat_Code_natural(semiri3763490453095760265atural,X),N) = aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).
% pochhammer_of_nat
tff(fact_3550_pochhammer__fact,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),N) ).
% pochhammer_fact
tff(fact_3551_pochhammer__fact,axiom,
! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_nat_Code_integer(comm_s8582702949713902594nteger(one_one_Code_integer),N) ).
% pochhammer_fact
tff(fact_3552_pochhammer__fact,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N) ).
% pochhammer_fact
tff(fact_3553_pochhammer__fact,axiom,
! [N: nat] : semiri2447717529341329178atural(N) = comm_s7406298101470620109atural(one_one_Code_natural,N) ).
% pochhammer_fact
tff(fact_3554_pochhammer__fact,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N) ).
% pochhammer_fact
tff(fact_3555_of__nat__gbinomial,axiom,
! [N: nat,K: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,N),K)) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) ).
% of_nat_gbinomial
tff(fact_3556_gbinomial__pochhammer,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_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_3557_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(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_3558_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).
% fact_mono_nat
tff(fact_3559_fact__ge__self,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_self
tff(fact_3560_power__not__zero,axiom,
! [A: code_integer,N: nat] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) != zero_z3403309356797280102nteger ) ) ).
% power_not_zero
tff(fact_3561_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).
% power_not_zero
tff(fact_3562_power__not__zero,axiom,
! [A: code_natural,N: nat] :
( ( A != zero_z2226904508553997617atural )
=> ( power_7079662738309270450atural(A,N) != zero_z2226904508553997617atural ) ) ).
% power_not_zero
tff(fact_3563_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).
% power_not_zero
tff(fact_3564_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).
% power_not_zero
tff(fact_3565_fact__nonzero,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).
% fact_nonzero
tff(fact_3566_fact__nonzero,axiom,
! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) != zero_z3403309356797280102nteger ).
% fact_nonzero
tff(fact_3567_fact__nonzero,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).
% fact_nonzero
tff(fact_3568_fact__nonzero,axiom,
! [N: nat] : semiri2447717529341329178atural(N) != zero_z2226904508553997617atural ).
% fact_nonzero
tff(fact_3569_fact__nonzero,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).
% fact_nonzero
tff(fact_3570_pochhammer__same,axiom,
! [N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% pochhammer_same
tff(fact_3571_pochhammer__same,axiom,
! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).
% pochhammer_same
tff(fact_3572_pochhammer__same,axiom,
! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% pochhammer_same
tff(fact_3573_Ints__power,axiom,
! [A: int,N: nat] :
( member_int(A,ring_1_Ints_int)
=> member_int(aa_nat_int(power_power_int(A),N),ring_1_Ints_int) ) ).
% Ints_power
tff(fact_3574_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(aa_rat_fun_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_3575_gbinomial__index__swap,axiom,
! [K: nat,N: nat] : aa_rat_rat(aa_rat_fun_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,N))),one_one_rat)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),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)),N)) ).
% gbinomial_index_swap
tff(fact_3576_gbinomial__negated__upper,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_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_3577_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).
% fact_less_mono_nat
tff(fact_3578_zero__le__power,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).
% zero_le_power
tff(fact_3579_zero__le__power,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),power_7079662738309270450atural(A,N))) ) ).
% zero_le_power
tff(fact_3580_zero__le__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% zero_le_power
tff(fact_3581_zero__le__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% zero_le_power
tff(fact_3582_zero__le__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).
% zero_le_power
tff(fact_3583_power__mono,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).
% power_mono
tff(fact_3584_power__mono,axiom,
! [A: code_natural,B: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(B,N))) ) ) ).
% power_mono
tff(fact_3585_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).
% power_mono
tff(fact_3586_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).
% power_mono
tff(fact_3587_power__mono,axiom,
! [A: int,B: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).
% power_mono
tff(fact_3588_zero__less__power,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),power_7079662738309270450atural(A,N))) ) ).
% zero_less_power
tff(fact_3589_zero__less__power,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).
% zero_less_power
tff(fact_3590_zero__less__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% zero_less_power
tff(fact_3591_zero__less__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% zero_less_power
tff(fact_3592_zero__less__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).
% zero_less_power
tff(fact_3593_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_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_3594_one__le__power,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).
% one_le_power
tff(fact_3595_one__le__power,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),power_7079662738309270450atural(A,N))) ) ).
% one_le_power
tff(fact_3596_one__le__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% one_le_power
tff(fact_3597_one__le__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% one_le_power
tff(fact_3598_one__le__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).
% one_le_power
tff(fact_3599_left__right__inverse__power,axiom,
! [X: code_natural,Y: code_natural,N: nat] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Y) = one_one_Code_natural )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(X,N)),power_7079662738309270450atural(Y,N)) = one_one_Code_natural ) ) ).
% left_right_inverse_power
tff(fact_3600_left__right__inverse__power,axiom,
! [X: code_integer,Y: code_integer,N: nat] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = one_one_Code_integer )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).
% left_right_inverse_power
tff(fact_3601_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N: nat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = one_one_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).
% left_right_inverse_power
tff(fact_3602_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).
% left_right_inverse_power
tff(fact_3603_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).
% left_right_inverse_power
tff(fact_3604_power__0,axiom,
! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).
% power_0
tff(fact_3605_power__0,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).
% power_0
tff(fact_3606_power__0,axiom,
! [A: code_natural] : power_7079662738309270450atural(A,zero_zero_nat) = one_one_Code_natural ).
% power_0
tff(fact_3607_power__0,axiom,
! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).
% power_0
tff(fact_3608_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_3609_power__one__over,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),N) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),N)) ).
% power_one_over
tff(fact_3610_power__add,axiom,
! [A: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_add
tff(fact_3611_power__add,axiom,
! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).
% power_add
tff(fact_3612_power__add,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_add
tff(fact_3613_power__add,axiom,
! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).
% power_add
tff(fact_3614_fact__ge__zero,axiom,
! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ).
% fact_ge_zero
tff(fact_3615_fact__ge__zero,axiom,
! [N: nat] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),semiri2447717529341329178atural(N))) ).
% fact_ge_zero
tff(fact_3616_fact__ge__zero,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).
% fact_ge_zero
tff(fact_3617_fact__ge__zero,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).
% fact_ge_zero
tff(fact_3618_fact__ge__zero,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_zero
tff(fact_3619_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% gbinomial_of_nat_symmetric
tff(fact_3620_fact__gt__zero,axiom,
! [N: nat] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),semiri2447717529341329178atural(N))) ).
% fact_gt_zero
tff(fact_3621_fact__gt__zero,axiom,
! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ).
% fact_gt_zero
tff(fact_3622_fact__gt__zero,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).
% fact_gt_zero
tff(fact_3623_fact__gt__zero,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).
% fact_gt_zero
tff(fact_3624_fact__gt__zero,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_gt_zero
tff(fact_3625_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,semiri2447717529341329178atural(N)),zero_z2226904508553997617atural)) ).
% fact_not_neg
tff(fact_3626_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),zero_z3403309356797280102nteger)) ).
% fact_not_neg
tff(fact_3627_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat)) ).
% fact_not_neg
tff(fact_3628_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).
% fact_not_neg
tff(fact_3629_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).
% fact_not_neg
tff(fact_3630_fact__ge__1,axiom,
! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ).
% fact_ge_1
tff(fact_3631_fact__ge__1,axiom,
! [N: nat] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),semiri2447717529341329178atural(N))) ).
% fact_ge_1
tff(fact_3632_fact__ge__1,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).
% fact_ge_1
tff(fact_3633_fact__ge__1,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).
% fact_ge_1
tff(fact_3634_fact__ge__1,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_1
tff(fact_3635_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).
% fact_mono
tff(fact_3636_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).
% fact_mono
tff(fact_3637_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).
% fact_mono
tff(fact_3638_pochhammer__pos,axiom,
! [X: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),X))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),comm_s7406298101470620109atural(X,N))) ) ).
% pochhammer_pos
tff(fact_3639_pochhammer__pos,axiom,
! [X: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(X),N))) ) ).
% pochhammer_pos
tff(fact_3640_pochhammer__pos,axiom,
! [X: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).
% pochhammer_pos
tff(fact_3641_pochhammer__pos,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).
% pochhammer_pos
tff(fact_3642_pochhammer__pos,axiom,
! [X: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).
% pochhammer_pos
tff(fact_3643_pochhammer__neq__0__mono,axiom,
! [A: rat,M: nat,N: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_3644_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_3645_gbinomial__minus,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).
% gbinomial_minus
tff(fact_3646_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_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_3647_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3648_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_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_3649_gbinomial__mult__1,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_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_3650_fact__ge__Suc__0__nat,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_Suc_0_nat
tff(fact_3651_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_3652_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(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ).
% pochhammer_minus
tff(fact_3653_pochhammer__minus,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3654_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_3655_pochhammer__minus_H,axiom,
! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_3656_pochhammer__minus_H,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3657_power__less__imp__less__base,axiom,
! [A: code_natural,N: nat,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(B,N)))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_3658_power__less__imp__less__base,axiom,
! [A: code_integer,N: nat,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_3659_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_3660_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_3661_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_3662_power__le__one,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer)) ) ) ).
% power_le_one
tff(fact_3663_power__le__one,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,N)),one_one_Code_natural)) ) ) ).
% power_le_one
tff(fact_3664_power__le__one,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).
% power_le_one
tff(fact_3665_power__le__one,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).
% power_le_one
tff(fact_3666_power__le__one,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).
% power_le_one
tff(fact_3667_power__le__imp__le__base,axiom,
! [A: code_integer,N: nat,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N))))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_3668_power__le__imp__le__base,axiom,
! [A: code_natural,N: nat,B: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,aa_nat_nat(suc,N))),power_7079662738309270450atural(B,aa_nat_nat(suc,N))))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_3669_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_3670_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_3671_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_3672_power__inject__base,axiom,
! [A: code_integer,N: nat,B: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N)) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_3673_power__inject__base,axiom,
! [A: code_natural,N: nat,B: code_natural] :
( ( power_7079662738309270450atural(A,aa_nat_nat(suc,N)) = power_7079662738309270450atural(B,aa_nat_nat(suc,N)) )
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_3674_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_3675_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_3676_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_3677_power__gt1__lemma,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N)))) ) ).
% power_gt1_lemma
tff(fact_3678_power__gt1__lemma,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))) ) ).
% power_gt1_lemma
tff(fact_3679_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).
% power_gt1_lemma
tff(fact_3680_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).
% power_gt1_lemma
tff(fact_3681_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).
% power_gt1_lemma
tff(fact_3682_power__less__power__Suc,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N)))) ) ).
% power_less_power_Suc
tff(fact_3683_power__less__power__Suc,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))) ) ).
% power_less_power_Suc
tff(fact_3684_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).
% power_less_power_Suc
tff(fact_3685_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).
% power_less_power_Suc
tff(fact_3686_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).
% power_less_power_Suc
tff(fact_3687_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = one_one_Code_integer ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ) ).
% power_0_left
tff(fact_3688_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).
% power_0_left
tff(fact_3689_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( power_7079662738309270450atural(zero_z2226904508553997617atural,N) = one_one_Code_natural ) )
& ( ( N != zero_zero_nat )
=> ( power_7079662738309270450atural(zero_z2226904508553997617atural,N) = zero_z2226904508553997617atural ) ) ) ).
% power_0_left
tff(fact_3690_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).
% power_0_left
tff(fact_3691_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).
% power_0_left
tff(fact_3692_power__gt1,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),power_7079662738309270450atural(A,aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_3693_power__gt1,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_3694_power__gt1,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_3695_power__gt1,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_3696_power__gt1,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_3697_power__less__imp__less__exp,axiom,
! [A: code_natural,M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,M)),power_7079662738309270450atural(A,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_3698_power__less__imp__less__exp,axiom,
! [A: code_integer,M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_3699_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_3700_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_3701_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_3702_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: code_natural] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(A,N4))) ) ) ).
% power_strict_increasing
tff(fact_3703_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N4))) ) ) ).
% power_strict_increasing
tff(fact_3704_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).
% power_strict_increasing
tff(fact_3705_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).
% power_strict_increasing
tff(fact_3706_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).
% power_strict_increasing
tff(fact_3707_power__minus,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).
% power_minus
tff(fact_3708_power__minus,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_minus
tff(fact_3709_power__minus,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).
% power_minus
tff(fact_3710_power__increasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N4))) ) ) ).
% power_increasing
tff(fact_3711_power__increasing,axiom,
! [N: nat,N4: nat,A: code_natural] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),A))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(A,N4))) ) ) ).
% power_increasing
tff(fact_3712_power__increasing,axiom,
! [N: nat,N4: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).
% power_increasing
tff(fact_3713_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).
% power_increasing
tff(fact_3714_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).
% power_increasing
tff(fact_3715_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ).
% zero_power
tff(fact_3716_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).
% zero_power
tff(fact_3717_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( power_7079662738309270450atural(zero_z2226904508553997617atural,N) = zero_z2226904508553997617atural ) ) ).
% zero_power
tff(fact_3718_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).
% zero_power
tff(fact_3719_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).
% zero_power
tff(fact_3720_zero__le__power__abs,axiom,
! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N))) ).
% zero_le_power_abs
tff(fact_3721_zero__le__power__abs,axiom,
! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N))) ).
% zero_le_power_abs
tff(fact_3722_zero__le__power__abs,axiom,
! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).
% zero_le_power_abs
tff(fact_3723_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ) ) ).
% fact_less_mono
tff(fact_3724_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ) ).
% fact_less_mono
tff(fact_3725_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).
% fact_less_mono
tff(fact_3726_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).
% fact_less_mono
tff(fact_3727_pochhammer__nonneg,axiom,
! [X: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),X))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),comm_s7406298101470620109atural(X,N))) ) ).
% pochhammer_nonneg
tff(fact_3728_pochhammer__nonneg,axiom,
! [X: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(X),N))) ) ).
% pochhammer_nonneg
tff(fact_3729_pochhammer__nonneg,axiom,
! [X: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).
% pochhammer_nonneg
tff(fact_3730_pochhammer__nonneg,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).
% pochhammer_nonneg
tff(fact_3731_pochhammer__nonneg,axiom,
! [X: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).
% pochhammer_nonneg
tff(fact_3732_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).
% pochhammer_0_left
tff(fact_3733_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).
% pochhammer_0_left
tff(fact_3734_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = one_one_Code_integer ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ) ).
% pochhammer_0_left
tff(fact_3735_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).
% pochhammer_0_left
tff(fact_3736_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( comm_s7406298101470620109atural(zero_z2226904508553997617atural,N) = one_one_Code_natural ) )
& ( ( N != zero_zero_nat )
=> ( comm_s7406298101470620109atural(zero_z2226904508553997617atural,N) = zero_z2226904508553997617atural ) ) ) ).
% pochhammer_0_left
tff(fact_3737_Suc__times__gbinomial,axiom,
! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).
% Suc_times_gbinomial
tff(fact_3738_gbinomial__absorption,axiom,
! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3739_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(aa_rat_fun_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,M),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_3740_power__Suc__less,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N))),power_7079662738309270450atural(A,N))) ) ) ).
% power_Suc_less
tff(fact_3741_power__Suc__less,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).
% power_Suc_less
tff(fact_3742_power__Suc__less,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).
% power_Suc_less
tff(fact_3743_power__Suc__less,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).
% power_Suc_less
tff(fact_3744_power__Suc__less,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).
% power_Suc_less
tff(fact_3745_power__Suc__le__self,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3746_power__Suc__le__self,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3747_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3748_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3749_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3750_power__Suc__less__one,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,aa_nat_nat(suc,N))),one_one_Code_natural)) ) ) ).
% power_Suc_less_one
tff(fact_3751_power__Suc__less__one,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),one_one_Code_integer)) ) ) ).
% power_Suc_less_one
tff(fact_3752_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).
% power_Suc_less_one
tff(fact_3753_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).
% power_Suc_less_one
tff(fact_3754_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).
% power_Suc_less_one
tff(fact_3755_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: code_natural] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,N4)),power_7079662738309270450atural(A,N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3756_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3757_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3758_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3759_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3760_power__decreasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ) ).
% power_decreasing
tff(fact_3761_power__decreasing,axiom,
! [N: nat,N4: nat,A: code_natural] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),one_one_Code_natural))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,N4)),power_7079662738309270450atural(A,N))) ) ) ) ).
% power_decreasing
tff(fact_3762_power__decreasing,axiom,
! [N: nat,N4: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).
% power_decreasing
tff(fact_3763_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).
% power_decreasing
tff(fact_3764_power__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).
% power_decreasing
tff(fact_3765_power__eq__imp__eq__base,axiom,
! [A: code_integer,N: nat,B: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_3766_power__eq__imp__eq__base,axiom,
! [A: code_natural,N: nat,B: code_natural] :
( ( power_7079662738309270450atural(A,N) = power_7079662738309270450atural(B,N) )
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_3767_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_3768_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_3769_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_3770_power__eq__iff__eq__base,axiom,
! [N: nat,A: code_integer,B: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_3771_power__eq__iff__eq__base,axiom,
! [N: nat,A: code_natural,B: code_natural] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B))
=> ( ( power_7079662738309270450atural(A,N) = power_7079662738309270450atural(B,N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_3772_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_3773_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_3774_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_3775_power__le__imp__le__exp,axiom,
! [A: code_natural,M: nat,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(A,M)),power_7079662738309270450atural(A,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_3776_power__le__imp__le__exp,axiom,
! [A: code_integer,M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_3777_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_3778_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_3779_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_3780_self__le__power,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).
% self_le_power
tff(fact_3781_self__le__power,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,A),power_7079662738309270450atural(A,N))) ) ) ).
% self_le_power
tff(fact_3782_self__le__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).
% self_le_power
tff(fact_3783_self__le__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).
% self_le_power
tff(fact_3784_self__le__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).
% self_le_power
tff(fact_3785_one__less__power,axiom,
! [A: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,one_one_Code_natural),power_7079662738309270450atural(A,N))) ) ) ).
% one_less_power
tff(fact_3786_one__less__power,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).
% one_less_power
tff(fact_3787_one__less__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).
% one_less_power
tff(fact_3788_one__less__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).
% one_less_power
tff(fact_3789_one__less__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).
% one_less_power
tff(fact_3790_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).
% power_diff
tff(fact_3791_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).
% power_diff
tff(fact_3792_power__diff,axiom,
! [A: code_integer,N: nat,M: nat] :
( ( A != zero_z3403309356797280102nteger )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).
% power_diff
tff(fact_3793_power__diff,axiom,
! [A: rat,N: nat,M: nat] :
( ( A != zero_zero_rat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).
% power_diff
tff(fact_3794_power__diff,axiom,
! [A: code_natural,N: nat,M: nat] :
( ( A != zero_z2226904508553997617atural )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,power_7079662738309270450atural(A,M)),power_7079662738309270450atural(A,N)) ) ) ) ).
% power_diff
tff(fact_3795_pochhammer__rec,axiom,
! [A: code_natural,N: nat] : comm_s7406298101470620109atural(A,aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),comm_s7406298101470620109atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural),N)) ).
% pochhammer_rec
tff(fact_3796_pochhammer__rec,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),N)) ).
% pochhammer_rec
tff(fact_3797_pochhammer__rec,axiom,
! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ).
% pochhammer_rec
tff(fact_3798_pochhammer__rec,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).
% pochhammer_rec
tff(fact_3799_pochhammer__rec,axiom,
! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ).
% pochhammer_rec
tff(fact_3800_pochhammer__Suc,axiom,
! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% pochhammer_Suc
tff(fact_3801_pochhammer__Suc,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).
% pochhammer_Suc
tff(fact_3802_pochhammer__Suc,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).
% pochhammer_Suc
tff(fact_3803_pochhammer__Suc,axiom,
! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).
% pochhammer_Suc
tff(fact_3804_pochhammer__Suc,axiom,
! [A: code_natural,N: nat] : comm_s7406298101470620109atural(A,aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,comm_s7406298101470620109atural(A,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_nat_Code_natural(semiri3763490453095760265atural,N))) ).
% pochhammer_Suc
tff(fact_3805_pochhammer__rec_H,axiom,
! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).
% pochhammer_rec'
tff(fact_3806_pochhammer__rec_H,axiom,
! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).
% pochhammer_rec'
tff(fact_3807_pochhammer__rec_H,axiom,
! [Z: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)) ).
% pochhammer_rec'
tff(fact_3808_pochhammer__rec_H,axiom,
! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).
% pochhammer_rec'
tff(fact_3809_pochhammer__rec_H,axiom,
! [Z: code_natural,N: nat] : comm_s7406298101470620109atural(Z,aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z),aa_nat_Code_natural(semiri3763490453095760265atural,N))),comm_s7406298101470620109atural(Z,N)) ).
% pochhammer_rec'
tff(fact_3810_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
<=> ? [K2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
& ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_3811_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_3812_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_3813_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_3814_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_3815_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_3816_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_3817_gbinomial__rec,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).
% gbinomial_rec
tff(fact_3818_gbinomial__factors,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_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_3819_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_3820_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_3821_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_3822_pochhammer__product_H,axiom,
! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).
% pochhammer_product'
tff(fact_3823_pochhammer__product_H,axiom,
! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).
% pochhammer_product'
tff(fact_3824_pochhammer__product_H,axiom,
! [Z: code_integer,N: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),M)) ).
% pochhammer_product'
tff(fact_3825_pochhammer__product_H,axiom,
! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).
% pochhammer_product'
tff(fact_3826_pochhammer__product_H,axiom,
! [Z: code_natural,N: nat,M: nat] : comm_s7406298101470620109atural(Z,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,comm_s7406298101470620109atural(Z,N)),comm_s7406298101470620109atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z),aa_nat_Code_natural(semiri3763490453095760265atural,N)),M)) ).
% pochhammer_product'
tff(fact_3827_power__minus_H,axiom,
! [X: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,one_one_int),X))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(X),N)) ) ) ).
% power_minus'
tff(fact_3828_power__minus_H,axiom,
! [X: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(nO_MAT8252062027627875367nteger,one_one_Code_integer),X))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).
% power_minus'
tff(fact_3829_power__minus_H,axiom,
! [X: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(nO_MATCH_rat_rat,one_one_rat),X))
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(X),N)) ) ) ).
% power_minus'
tff(fact_3830_power__strict__mono,axiom,
! [A: code_natural,B: code_natural,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,A),B))
=> ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(B,N))) ) ) ) ).
% power_strict_mono
tff(fact_3831_power__strict__mono,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ) ).
% power_strict_mono
tff(fact_3832_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).
% power_strict_mono
tff(fact_3833_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).
% power_strict_mono
tff(fact_3834_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).
% power_strict_mono
tff(fact_3835_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).
% power_diff_power_eq
tff(fact_3836_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).
% power_diff_power_eq
tff(fact_3837_power__diff__power__eq,axiom,
! [A: code_integer,N: nat,M: nat] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).
% power_diff_power_eq
tff(fact_3838_power__diff__power__eq,axiom,
! [A: code_natural,N: nat,M: nat] :
( ( A != zero_z2226904508553997617atural )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,power_7079662738309270450atural(A,M)),power_7079662738309270450atural(A,N)) = power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,power_7079662738309270450atural(A,M)),power_7079662738309270450atural(A,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,one_one_Code_natural),power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).
% power_diff_power_eq
tff(fact_3839_power__eq__if,axiom,
! [M: nat,P3: code_natural] :
( ( ( M = zero_zero_nat )
=> ( power_7079662738309270450atural(P3,M) = one_one_Code_natural ) )
& ( ( M != zero_zero_nat )
=> ( power_7079662738309270450atural(P3,M) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,P3),power_7079662738309270450atural(P3,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3840_power__eq__if,axiom,
! [M: nat,P3: code_integer] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = one_one_Code_integer ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P3),aa_nat_Code_integer(power_8256067586552552935nteger(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3841_power__eq__if,axiom,
! [M: nat,P3: rat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(P3),M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3842_power__eq__if,axiom,
! [M: nat,P3: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3843_power__eq__if,axiom,
! [M: nat,P3: int] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_int(power_power_int(P3),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3844_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3845_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3846_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3847_power__minus__mult,axiom,
! [N: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).
% power_minus_mult
tff(fact_3848_power__minus__mult,axiom,
! [N: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).
% power_minus_mult
tff(fact_3849_power__minus__mult,axiom,
! [N: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).
% power_minus_mult
tff(fact_3850_power__minus__mult,axiom,
! [N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).
% power_minus_mult
tff(fact_3851_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_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_3852_pochhammer__product,axiom,
! [M: nat,N: nat,Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_3853_pochhammer__product,axiom,
! [M: nat,N: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_3854_pochhammer__product,axiom,
! [M: nat,N: nat,Z: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_3855_pochhammer__product,axiom,
! [M: nat,N: nat,Z: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_3856_pochhammer__product,axiom,
! [M: nat,N: nat,Z: code_natural] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( comm_s7406298101470620109atural(Z,N) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,comm_s7406298101470620109atural(Z,M)),comm_s7406298101470620109atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z),aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_3857_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_3858_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_Code_integer(semiri3624122377584611663nteger,M) = one_one_Code_integer ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_Code_integer(semiri3624122377584611663nteger,M) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_3859_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_rat(semiri773545260158071498ct_rat,M) = one_one_rat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_rat(semiri773545260158071498ct_rat,M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_3860_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( semiri2447717529341329178atural(M) = one_one_Code_natural ) )
& ( ( M != zero_zero_nat )
=> ( semiri2447717529341329178atural(M) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),semiri2447717529341329178atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_3861_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_3862_gbinomial__code,axiom,
! [K: nat,A: rat] :
( ( ( K = zero_zero_nat )
=> ( aa_nat_rat(gbinomial_rat(A),K) = one_one_rat ) )
& ( ( K != zero_zero_nat )
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_bv(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_3863_pochhammer__code,axiom,
! [N: nat,A: int] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = set_fo2581907887559384638at_int(aTP_Lamm_bw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_int) ) ) ) ).
% pochhammer_code
tff(fact_3864_pochhammer__code,axiom,
! [N: nat,A: code_integer] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = one_one_Code_integer ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_bx(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_Code_integer) ) ) ) ).
% pochhammer_code
tff(fact_3865_pochhammer__code,axiom,
! [N: nat,A: rat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_by(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).
% pochhammer_code
tff(fact_3866_pochhammer__code,axiom,
! [N: nat,A: code_natural] :
( ( ( N = zero_zero_nat )
=> ( comm_s7406298101470620109atural(A,N) = one_one_Code_natural ) )
& ( ( N != zero_zero_nat )
=> ( comm_s7406298101470620109atural(A,N) = set_fo9131927060563008058atural(aTP_Lamm_bz(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_Code_natural) ) ) ) ).
% pochhammer_code
tff(fact_3867_pochhammer__code,axiom,
! [N: nat,A: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ca(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).
% pochhammer_code
tff(fact_3868_slice__nth,axiom,
! [From: nat,To: nat,Xs: list_nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
=> ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).
% slice_nth
tff(fact_3869_slice__nth,axiom,
! [From: nat,To: nat,Xs: list_int,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_int_nat(size_size_list_int,Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
=> ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).
% slice_nth
tff(fact_3870_slice__nth,axiom,
! [From: nat,To: nat,Xs: list_bool,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_bool_nat(size_size_list_bool,Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
=> ( pp(aa_nat_bool(nth_bool(slice_bool(From,To,Xs)),I))
<=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I))) ) ) ) ) ).
% slice_nth
tff(fact_3871_split__pos__lemma,axiom,
! [K: int,P: fun_int_fun_int_bool,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
<=> ! [I3: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).
% split_pos_lemma
tff(fact_3872_split__neg__lemma,axiom,
! [K: int,P: fun_int_fun_int_bool,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
<=> ! [I3: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).
% split_neg_lemma
tff(fact_3873_verit__le__mono__div__int,axiom,
! [A3: int,B3: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N)),zero_zero_int),one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N))) ) ) ).
% verit_le_mono_div_int
tff(fact_3874_verit__le__mono__div,axiom,
! [A3: nat,B3: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N))) ) ) ).
% verit_le_mono_div
tff(fact_3875_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_3876_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_3877_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_3878_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_num_Code_integer(numera6620942414471956472nteger,M) = aa_num_Code_integer(numera6620942414471956472nteger,N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_3879_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_num_rat(numeral_numeral_rat,M) = aa_num_rat(numeral_numeral_rat,N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_3880_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( numera5444537566228673987atural(M) = numera5444537566228673987atural(N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_3881_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
<=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).
% int_eq_iff_numeral
tff(fact_3882_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_3883_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_3884_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,numera5444537566228673987atural(M)),numera5444537566228673987atural(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_3885_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_3886_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_3887_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_3888_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_3889_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_3890_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_3891_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_3892_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,numera5444537566228673987atural(M)),numera5444537566228673987atural(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_3893_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).
% numeral_plus_numeral
tff(fact_3894_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)) ).
% numeral_plus_numeral
tff(fact_3895_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),N)) ).
% numeral_plus_numeral
tff(fact_3896_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).
% numeral_plus_numeral
tff(fact_3897_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(M)),numera5444537566228673987atural(N)) = numera5444537566228673987atural(aa_num_num(plus_plus_num(M),N)) ).
% numeral_plus_numeral
tff(fact_3898_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(plus_plus_num(V),W))),Z) ).
% add_numeral_left
tff(fact_3899_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(plus_plus_num(V),W))),Z) ).
% add_numeral_left
tff(fact_3900_add__numeral__left,axiom,
! [V: num,W: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W))),Z) ).
% add_numeral_left
tff(fact_3901_add__numeral__left,axiom,
! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Z) ).
% add_numeral_left
tff(fact_3902_add__numeral__left,axiom,
! [V: num,W: num,Z: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(V)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(W)),Z)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(aa_num_num(plus_plus_num(V),W))),Z) ).
% add_numeral_left
tff(fact_3903_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).
% numeral_times_numeral
tff(fact_3904_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N)) ).
% numeral_times_numeral
tff(fact_3905_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N)) ).
% numeral_times_numeral
tff(fact_3906_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).
% numeral_times_numeral
tff(fact_3907_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(M)),numera5444537566228673987atural(N)) = numera5444537566228673987atural(aa_num_num(times_times_num(M),N)) ).
% numeral_times_numeral
tff(fact_3908_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(times_times_num(V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3909_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(times_times_num(V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3910_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3911_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3912_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(W)),Z)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(aa_num_num(times_times_num(V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3913_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_3914_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_3915_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_3916_power__zero__numeral,axiom,
! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).
% power_zero_numeral
tff(fact_3917_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_3918_power__zero__numeral,axiom,
! [K: num] : power_7079662738309270450atural(zero_z2226904508553997617atural,aa_num_nat(numeral_numeral_nat,K)) = zero_z2226904508553997617atural ).
% power_zero_numeral
tff(fact_3919_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_3920_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_3921_mod__self,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).
% mod_self
tff(fact_3922_mod__self,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).
% mod_self
tff(fact_3923_mod__self,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),A) = zero_z3403309356797280102nteger ).
% mod_self
tff(fact_3924_mod__self,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),A) = zero_z2226904508553997617atural ).
% mod_self
tff(fact_3925_mod__by__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).
% mod_by_0
tff(fact_3926_mod__by__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).
% mod_by_0
tff(fact_3927_mod__by__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),zero_z3403309356797280102nteger) = A ).
% mod_by_0
tff(fact_3928_mod__by__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),zero_z2226904508553997617atural) = A ).
% mod_by_0
tff(fact_3929_mod__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).
% mod_0
tff(fact_3930_mod__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).
% mod_0
tff(fact_3931_mod__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).
% mod_0
tff(fact_3932_mod__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).
% mod_0
tff(fact_3933_bits__mod__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).
% bits_mod_0
tff(fact_3934_bits__mod__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).
% bits_mod_0
tff(fact_3935_bits__mod__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).
% bits_mod_0
tff(fact_3936_bits__mod__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).
% bits_mod_0
tff(fact_3937_power__add__numeral,axiom,
! [A: code_integer,M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).
% power_add_numeral
tff(fact_3938_power__add__numeral,axiom,
! [A: rat,M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).
% power_add_numeral
tff(fact_3939_power__add__numeral,axiom,
! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).
% power_add_numeral
tff(fact_3940_power__add__numeral,axiom,
! [A: int,M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N))) ).
% power_add_numeral
tff(fact_3941_power__add__numeral2,axiom,
! [A: code_integer,M: num,N: num,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B) ).
% power_add_numeral2
tff(fact_3942_power__add__numeral2,axiom,
! [A: rat,M: num,N: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B) ).
% power_add_numeral2
tff(fact_3943_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B) ).
% power_add_numeral2
tff(fact_3944_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)))),B) ).
% power_add_numeral2
tff(fact_3945_of__nat__numeral,axiom,
! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).
% of_nat_numeral
tff(fact_3946_of__nat__numeral,axiom,
! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).
% of_nat_numeral
tff(fact_3947_of__nat__numeral,axiom,
! [N: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,N)) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).
% of_nat_numeral
tff(fact_3948_of__nat__numeral,axiom,
! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_rat(numeral_numeral_rat,N) ).
% of_nat_numeral
tff(fact_3949_of__nat__numeral,axiom,
! [N: num] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_num_nat(numeral_numeral_nat,N)) = numera5444537566228673987atural(N) ).
% of_nat_numeral
tff(fact_3950_mod__add__self1,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_add_self1
tff(fact_3951_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_3952_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).
% mod_add_self1
tff(fact_3953_mod__add__self1,axiom,
! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) ).
% mod_add_self1
tff(fact_3954_mod__add__self2,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_add_self2
tff(fact_3955_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_3956_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).
% mod_add_self2
tff(fact_3957_mod__add__self2,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) ).
% mod_add_self2
tff(fact_3958_abs__numeral,axiom,
! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).
% abs_numeral
tff(fact_3959_abs__numeral,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).
% abs_numeral
tff(fact_3960_abs__numeral,axiom,
! [N: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,N) ).
% abs_numeral
tff(fact_3961_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_3962_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = aa_num_Code_integer(numera6620942414471956472nteger,N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_3963_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_3964_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_3965_of__int__numeral,axiom,
! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_num_int(numeral_numeral_int,K)) = aa_num_Code_integer(numera6620942414471956472nteger,K) ).
% of_int_numeral
tff(fact_3966_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_3967_power__int__add__numeral,axiom,
! [X: rat,M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,M))),aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,N))) = aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N))) ).
% power_int_add_numeral
tff(fact_3968_power__int__add__numeral2,axiom,
! [X: rat,M: num,N: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,M))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,N))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)))),B) ).
% power_int_add_numeral2
tff(fact_3969_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_3970_ceiling__numeral,axiom,
! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).
% ceiling_numeral
tff(fact_3971_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_3972_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_3973_distrib__right__numeral,axiom,
! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_num_Code_integer(numera6620942414471956472nteger,V))) ).
% distrib_right_numeral
tff(fact_3974_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).
% distrib_right_numeral
tff(fact_3975_distrib__right__numeral,axiom,
! [A: code_natural,B: code_natural,V: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),numera5444537566228673987atural(V)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),numera5444537566228673987atural(V))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),numera5444537566228673987atural(V))) ).
% distrib_right_numeral
tff(fact_3976_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_3977_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_3978_distrib__left__numeral,axiom,
! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ).
% distrib_left_numeral
tff(fact_3979_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C)) ).
% distrib_left_numeral
tff(fact_3980_distrib__left__numeral,axiom,
! [V: num,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V)),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V)),C)) ).
% distrib_left_numeral
tff(fact_3981_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_3982_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_3983_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_3984_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_3985_left__diff__distrib__numeral,axiom,
! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_num_Code_integer(numera6620942414471956472nteger,V))) ).
% left_diff_distrib_numeral
tff(fact_3986_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).
% left_diff_distrib_numeral
tff(fact_3987_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_3988_right__diff__distrib__numeral,axiom,
! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ).
% right_diff_distrib_numeral
tff(fact_3989_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C)) ).
% right_diff_distrib_numeral
tff(fact_3990_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_3991_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_3992_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_3993_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(plus_plus_num(V),W)))),Y) ).
% semiring_norm(168)
tff(fact_3994_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(V),W)))),Y) ).
% semiring_norm(168)
tff(fact_3995_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W)))),Y) ).
% semiring_norm(168)
tff(fact_3996_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).
% add_neg_numeral_simps(3)
tff(fact_3997_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% add_neg_numeral_simps(3)
tff(fact_3998_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N))) ).
% add_neg_numeral_simps(3)
tff(fact_3999_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4000_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4001_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4002_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4003_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4004_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4005_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4006_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(times_times_num(M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4007_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4008_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N))) ).
% diff_numeral_simps(3)
tff(fact_4009_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),N))) ).
% diff_numeral_simps(3)
tff(fact_4010_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N))) ).
% diff_numeral_simps(3)
tff(fact_4011_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).
% diff_numeral_simps(2)
tff(fact_4012_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(plus_plus_num(M),N)) ).
% diff_numeral_simps(2)
tff(fact_4013_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).
% diff_numeral_simps(2)
tff(fact_4014_mod__mult__self2__is__0,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).
% mod_mult_self2_is_0
tff(fact_4015_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_4016_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = zero_z3403309356797280102nteger ).
% mod_mult_self2_is_0
tff(fact_4017_mod__mult__self2__is__0,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),B) = zero_z2226904508553997617atural ).
% mod_mult_self2_is_0
tff(fact_4018_mod__mult__self1__is__0,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).
% mod_mult_self1_is_0
tff(fact_4019_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_4020_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),B) = zero_z3403309356797280102nteger ).
% mod_mult_self1_is_0
tff(fact_4021_mod__mult__self1__is__0,axiom,
! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)),B) = zero_z2226904508553997617atural ).
% mod_mult_self1_is_0
tff(fact_4022_mod__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).
% mod_by_1
tff(fact_4023_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_4024_mod__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% mod_by_1
tff(fact_4025_mod__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),one_one_Code_natural) = zero_z2226904508553997617atural ).
% mod_by_1
tff(fact_4026_bits__mod__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).
% bits_mod_by_1
tff(fact_4027_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_4028_bits__mod__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% bits_mod_by_1
tff(fact_4029_bits__mod__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),one_one_Code_natural) = zero_z2226904508553997617atural ).
% bits_mod_by_1
tff(fact_4030_mod__mult__self1,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self1
tff(fact_4031_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_4032_mod__mult__self1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).
% mod_mult_self1
tff(fact_4033_mod__mult__self1,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) ).
% mod_mult_self1
tff(fact_4034_mod__mult__self2,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self2
tff(fact_4035_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_4036_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).
% mod_mult_self2
tff(fact_4037_mod__mult__self2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) ).
% mod_mult_self2
tff(fact_4038_mod__mult__self3,axiom,
! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self3
tff(fact_4039_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_4040_mod__mult__self3,axiom,
! [C: code_integer,B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).
% mod_mult_self3
tff(fact_4041_mod__mult__self3,axiom,
! [C: code_natural,B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)),A)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) ).
% mod_mult_self3
tff(fact_4042_mod__mult__self4,axiom,
! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self4
tff(fact_4043_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_4044_mod__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) ).
% mod_mult_self4
tff(fact_4045_mod__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),A)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) ).
% mod_mult_self4
tff(fact_4046_mod__div__trivial,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).
% mod_div_trivial
tff(fact_4047_mod__div__trivial,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).
% mod_div_trivial
tff(fact_4048_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),B) = zero_z3403309356797280102nteger ).
% mod_div_trivial
tff(fact_4049_mod__div__trivial,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)),B) = zero_z2226904508553997617atural ).
% mod_div_trivial
tff(fact_4050_bits__mod__div__trivial,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).
% bits_mod_div_trivial
tff(fact_4051_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).
% bits_mod_div_trivial
tff(fact_4052_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),B) = zero_z3403309356797280102nteger ).
% bits_mod_div_trivial
tff(fact_4053_bits__mod__div__trivial,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)),B) = zero_z2226904508553997617atural ).
% bits_mod_div_trivial
tff(fact_4054_abs__neg__numeral,axiom,
! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).
% abs_neg_numeral
tff(fact_4055_abs__neg__numeral,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).
% abs_neg_numeral
tff(fact_4056_abs__neg__numeral,axiom,
! [N: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,N) ).
% abs_neg_numeral
tff(fact_4057_power__int__mult__distrib__numeral1,axiom,
! [W: num,Y: rat,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Y)),M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(aa_num_rat(numeral_numeral_rat,W)),M)),aa_int_rat(power_int_rat(Y),M)) ).
% power_int_mult_distrib_numeral1
tff(fact_4058_power__int__mult__distrib__numeral2,axiom,
! [X: rat,W: num,M: int] : aa_int_rat(power_int_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_num_rat(numeral_numeral_rat,W))),M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(power_int_rat(X),M)),aa_int_rat(power_int_rat(aa_num_rat(numeral_numeral_rat,W)),M)) ).
% power_int_mult_distrib_numeral2
tff(fact_4059_mod__by__Suc__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
% mod_by_Suc_0
tff(fact_4060_nat__neg__numeral,axiom,
! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).
% nat_neg_numeral
tff(fact_4061_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( M = zero_zero_nat )
| ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% nat_power_eq_Suc_0_iff
tff(fact_4062_power__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).
% power_Suc_0
tff(fact_4063_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
tff(fact_4064_diff__nat__numeral,axiom,
! [V: num,V2: 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,V2)) = 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,V2))) ).
% diff_nat_numeral
tff(fact_4065_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% nat_eq_numeral_power_cancel_iff
tff(fact_4066_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_nat_cancel_iff
tff(fact_4067_dbl__simps_I1_J,axiom,
! [K: num] : neg_numeral_dbl_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,neg_numeral_dbl_int(aa_num_int(numeral_numeral_int,K))) ).
% dbl_simps(1)
tff(fact_4068_dbl__simps_I1_J,axiom,
! [K: num] : aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) ).
% dbl_simps(1)
tff(fact_4069_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_4070_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B)) ) ).
% le_divide_eq_numeral1(1)
tff(fact_4071_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)))) ) ).
% divide_le_eq_numeral1(1)
tff(fact_4072_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)) )
<=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)) = B ) )
& ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_4073_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 )
<=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)) ) )
& ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_4074_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)))) ) ).
% divide_less_eq_numeral1(1)
tff(fact_4075_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B)) ) ).
% less_divide_eq_numeral1(1)
tff(fact_4076_mod__minus1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% mod_minus1_right
tff(fact_4077_mod__minus1__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% mod_minus1_right
tff(fact_4078_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_le_iff
tff(fact_4079_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_le_iff
tff(fact_4080_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_le_iff
tff(fact_4081_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_le_numeral_iff
tff(fact_4082_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_le_numeral_iff
tff(fact_4083_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_le_numeral_iff
tff(fact_4084_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_less_iff
tff(fact_4085_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_less_iff
tff(fact_4086_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_less_iff
tff(fact_4087_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_less_numeral_iff
tff(fact_4088_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_less_numeral_iff
tff(fact_4089_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_less_numeral_iff
tff(fact_4090_numeral__le__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).
% numeral_le_floor
tff(fact_4091_floor__less__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).
% floor_less_numeral
tff(fact_4092_ceiling__le__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).
% ceiling_le_numeral
tff(fact_4093_numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).
% numeral_less_ceiling
tff(fact_4094_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_4095_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_4096_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_4097_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_4098_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( power_7079662738309270450atural(numera5444537566228673987atural(X),N) = aa_nat_Code_natural(semiri3763490453095760265atural,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_4099_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4100_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4101_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4102_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4103_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_Code_natural(semiri3763490453095760265atural,Y) = power_7079662738309270450atural(numera5444537566228673987atural(X),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4104_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self1
tff(fact_4105_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self2
tff(fact_4106_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self3
tff(fact_4107_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self4
tff(fact_4108_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_4109_ceiling__add__numeral,axiom,
! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_add_numeral
tff(fact_4110_floor__diff__numeral,axiom,
! [X: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)) ).
% floor_diff_numeral
tff(fact_4111_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_4112_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
tff(fact_4113_ceiling__diff__numeral,axiom,
! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_diff_numeral
tff(fact_4114_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_4115_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_4116_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_4117_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_int_int(ring_1_of_int_int,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_4118_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_4119_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_4120_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).
% mod_neg_neg_trivial
tff(fact_4121_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).
% mod_pos_pos_trivial
tff(fact_4122_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_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_4123_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_4124_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))) )
<=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B ) )
& ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_4125_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 )
<=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat )
=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ) )
& ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_4126_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_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_4127_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_4128_dbl__dec__simps_I1_J,axiom,
! [K: num] : neg_nu3811975205180677377ec_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,neg_nu5851722552734809277nc_int(aa_num_int(numeral_numeral_int,K))) ).
% dbl_dec_simps(1)
tff(fact_4129_dbl__dec__simps_I1_J,axiom,
! [K: num] : aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) ).
% dbl_dec_simps(1)
tff(fact_4130_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_4131_dbl__inc__simps_I1_J,axiom,
! [K: num] : neg_nu5851722552734809277nc_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,neg_nu3811975205180677377ec_int(aa_num_int(numeral_numeral_int,K))) ).
% dbl_inc_simps(1)
tff(fact_4132_dbl__inc__simps_I1_J,axiom,
! [K: num] : aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) ).
% dbl_inc_simps(1)
tff(fact_4133_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_4134_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_4135_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_nat_cancel_iff
tff(fact_4136_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% nat_less_numeral_power_cancel_iff
tff(fact_4137_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% nat_le_numeral_power_cancel_iff
tff(fact_4138_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_nat_cancel_iff
tff(fact_4139_numeral__less__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).
% numeral_less_floor
tff(fact_4140_floor__le__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).
% floor_le_numeral
tff(fact_4141_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).
% ceiling_less_numeral
tff(fact_4142_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).
% numeral_le_ceiling
tff(fact_4143_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_4144_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_4145_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_4146_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_4147_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,power_7079662738309270450atural(numera5444537566228673987atural(I),N)),aa_nat_Code_natural(semiri3763490453095760265atural,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_4148_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_4149_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_4150_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_4151_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_4152_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),power_7079662738309270450atural(numera5444537566228673987atural(I),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_4153_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_4154_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),power_7079662738309270450atural(numera5444537566228673987atural(I),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_4155_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_4156_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_4157_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_4158_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_4159_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,power_7079662738309270450atural(numera5444537566228673987atural(I),N)),aa_nat_Code_natural(semiri3763490453095760265atural,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_4160_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_4161_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_4162_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_4163_neg__numeral__le__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).
% neg_numeral_le_floor
tff(fact_4164_floor__less__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).
% floor_less_neg_numeral
tff(fact_4165_ceiling__le__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).
% ceiling_le_neg_numeral
tff(fact_4166_neg__numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).
% neg_numeral_less_ceiling
tff(fact_4167_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_4168_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_4169_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_4170_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_4171_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_4172_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_4173_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_4174_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_4175_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_4176_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_4177_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_4178_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_4179_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4180_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4181_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4182_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = aa_int_int(ring_1_of_int_int,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4183_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4184_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4185_neg__numeral__less__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).
% neg_numeral_less_floor
tff(fact_4186_floor__le__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_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_4187_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),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_4188_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(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)),X)) ) ).
% neg_numeral_le_ceiling
tff(fact_4189_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4190_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4191_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4192_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4193_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4194_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4195_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4196_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4197_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4198_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4199_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4200_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4201_zmod__int,axiom,
! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% zmod_int
tff(fact_4202_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_4203_int__ops_I3_J,axiom,
! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).
% int_ops(3)
tff(fact_4204_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_mod
tff(fact_4205_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_mod
tff(fact_4206_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% of_nat_mod
tff(fact_4207_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% of_nat_mod
tff(fact_4208_mod__add__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_eq
tff(fact_4209_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_4210_mod__add__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).
% mod_add_eq
tff(fact_4211_mod__add__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,B),C))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) ).
% mod_add_eq
tff(fact_4212_mod__add__cong,axiom,
! [A: int,C: int,A4: int,B: int,B4: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B4)),C) ) ) ) ).
% mod_add_cong
tff(fact_4213_mod__add__cong,axiom,
! [A: nat,C: nat,A4: nat,B: nat,B4: 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,A4),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,B4),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,A4),B4)),C) ) ) ) ).
% mod_add_cong
tff(fact_4214_mod__add__cong,axiom,
! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A4),C) )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B4),C) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A4),B4)),C) ) ) ) ).
% mod_add_cong
tff(fact_4215_mod__add__cong,axiom,
! [A: code_natural,C: code_natural,A4: code_natural,B: code_natural,B4: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A4),C) )
=> ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,B4),C) )
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A4),B4)),C) ) ) ) ).
% mod_add_cong
tff(fact_4216_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_left_eq
tff(fact_4217_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_4218_mod__add__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C)),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).
% mod_add_left_eq
tff(fact_4219_mod__add__left__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),C)),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) ).
% mod_add_left_eq
tff(fact_4220_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_right_eq
tff(fact_4221_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_4222_mod__add__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).
% mod_add_right_eq
tff(fact_4223_mod__add__right__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,B),C))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) ).
% mod_add_right_eq
tff(fact_4224_zero__neq__numeral,axiom,
! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).
% zero_neq_numeral
tff(fact_4225_zero__neq__numeral,axiom,
! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).
% zero_neq_numeral
tff(fact_4226_zero__neq__numeral,axiom,
! [N: num] : zero_z3403309356797280102nteger != aa_num_Code_integer(numera6620942414471956472nteger,N) ).
% zero_neq_numeral
tff(fact_4227_zero__neq__numeral,axiom,
! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).
% zero_neq_numeral
tff(fact_4228_zero__neq__numeral,axiom,
! [N: num] : zero_z2226904508553997617atural != numera5444537566228673987atural(N) ).
% zero_neq_numeral
tff(fact_4229_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).
% mod_less_eq_dividend
tff(fact_4230_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).
% neg_numeral_neq_numeral
tff(fact_4231_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) != aa_num_Code_integer(numera6620942414471956472nteger,N) ).
% neg_numeral_neq_numeral
tff(fact_4232_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,N) ).
% neg_numeral_neq_numeral
tff(fact_4233_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% numeral_neq_neg_numeral
tff(fact_4234_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : aa_num_Code_integer(numera6620942414471956472nteger,M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).
% numeral_neq_neg_numeral
tff(fact_4235_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).
% numeral_neq_neg_numeral
tff(fact_4236_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(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_mod_as_int
tff(fact_4237_Ints__numeral,axiom,
! [N: num] : member_int(aa_num_int(numeral_numeral_int,N),ring_1_Ints_int) ).
% Ints_numeral
tff(fact_4238_Ints__numeral,axiom,
! [N: num] : member_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,N),ring_11222124179247155820nteger) ).
% Ints_numeral
tff(fact_4239_Ints__numeral,axiom,
! [N: num] : member_rat(aa_num_rat(numeral_numeral_rat,N),ring_1_Ints_rat) ).
% Ints_numeral
tff(fact_4240_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).
% nat_mod_distrib
tff(fact_4241_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).
% mod_abs_eq_div_nat
tff(fact_4242_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_4243_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,aa_num_Code_integer(numera6620942414471956472nteger,K)) ).
% of_int_neg_numeral
tff(fact_4244_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_4245_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),A)) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4246_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A)) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4247_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A)) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4248_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4249_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4250_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4251_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
=> ~ ! [D5: 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),D5)) ) ).
% mod_eqE
tff(fact_4252_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C) )
=> ~ ! [D5: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D5)) ) ).
% mod_eqE
tff(fact_4253_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
<=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_4254_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_4255_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = A )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B) = zero_z3403309356797280102nteger ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_4256_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_natural,B: code_natural] :
( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B) = A )
<=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B) = zero_z2226904508553997617atural ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_4257_div__add1__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).
% div_add1_eq
tff(fact_4258_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_4259_div__add1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C)) ).
% div_add1_eq
tff(fact_4260_div__add1__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,B),C))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,B),C))),C)) ).
% div_add1_eq
tff(fact_4261_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
=> ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).
% nth_rotate1
tff(fact_4262_nth__rotate1,axiom,
! [N: nat,Xs: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
=> ( aa_nat_int(nth_int(rotate1_int(Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),aa_list_int_nat(size_size_list_int,Xs))) ) ) ).
% nth_rotate1
tff(fact_4263_nth__rotate1,axiom,
! [N: nat,Xs: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
=> ( pp(aa_nat_bool(nth_bool(rotate1_bool(Xs)),N))
<=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),aa_list_bool_nat(size_size_list_bool,Xs)))) ) ) ).
% nth_rotate1
tff(fact_4264_list__eq__iff__nth__eq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
<=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
=> ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_4265_list__eq__iff__nth__eq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
<=> ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
=> ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_4266_list__eq__iff__nth__eq,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( Xs = Ys )
<=> ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,Xs)))
=> ( pp(aa_nat_bool(nth_bool(Xs),I3))
<=> pp(aa_nat_bool(nth_bool(Ys),I3)) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_4267_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_nat_bool] :
( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
=> ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),X_13)) )
<=> ? [Xs3: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(nth_nat(Xs3),I3))) ) ) ) ).
% Skolem_list_nth
tff(fact_4268_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_int_bool] :
( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
=> ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),X_13)) )
<=> ? [Xs3: list_int] :
( ( aa_list_int_nat(size_size_list_int,Xs3) = K )
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
=> pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),aa_nat_int(nth_int(Xs3),I3))) ) ) ) ).
% Skolem_list_nth
tff(fact_4269_Skolem__list__nth,axiom,
! [K: nat,P: fun_na9137082442855928016l_bool] :
( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
=> ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),X_13)) )
<=> ? [Xs3: list_bool] :
( ( aa_list_bool_nat(size_size_list_bool,Xs3) = K )
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
=> pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),aa_nat_bool(nth_bool(Xs3),I3))) ) ) ) ).
% Skolem_list_nth
tff(fact_4270_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
=> ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_4271_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs)))
=> ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_4272_nth__equalityI,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_bool_nat(size_size_list_bool,Xs)))
=> ( pp(aa_nat_bool(nth_bool(Xs),I2))
<=> pp(aa_nat_bool(nth_bool(Ys),I2)) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_4273_obtain__list__from__elements,axiom,
! [N: nat,P: fun_nat_fun_nat_bool] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> ? [Li: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,Li),I2)) )
=> ~ ! [L2: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,L2) = N )
=> ~ ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(L2),I4)),I4)) ) ) ) ).
% obtain_list_from_elements
tff(fact_4274_obtain__list__from__elements,axiom,
! [N: nat,P: fun_int_fun_nat_bool] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> ? [Li: int] : pp(aa_nat_bool(aa_int_fun_nat_bool(P,Li),I2)) )
=> ~ ! [L2: list_int] :
( ( aa_list_int_nat(size_size_list_int,L2) = N )
=> ~ ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
=> pp(aa_nat_bool(aa_int_fun_nat_bool(P,aa_nat_int(nth_int(L2),I4)),I4)) ) ) ) ).
% obtain_list_from_elements
tff(fact_4275_obtain__list__from__elements,axiom,
! [N: nat,P: fun_bo3288154511880838028t_bool] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> ? [Li: bool] : pp(aa_nat_bool(aa_bool_fun_nat_bool(P,Li),I2)) )
=> ~ ! [L2: list_bool] :
( ( aa_list_bool_nat(size_size_list_bool,L2) = N )
=> ~ ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
=> pp(aa_nat_bool(aa_bool_fun_nat_bool(P,aa_nat_bool(nth_bool(L2),I4)),I4)) ) ) ) ).
% obtain_list_from_elements
tff(fact_4276_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
& ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).
% mod_Suc
tff(fact_4277_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).
% mod_less_divisor
tff(fact_4278_gcd__nat__induct,axiom,
! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
( ! [M3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),zero_zero_nat))
=> ( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),N2)) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).
% gcd_nat_induct
tff(fact_4279_zero__le__numeral,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% zero_le_numeral
tff(fact_4280_zero__le__numeral,axiom,
! [N: num] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,zero_z2226904508553997617atural),numera5444537566228673987atural(N))) ).
% zero_le_numeral
tff(fact_4281_zero__le__numeral,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).
% zero_le_numeral
tff(fact_4282_zero__le__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).
% zero_le_numeral
tff(fact_4283_zero__le__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).
% zero_le_numeral
tff(fact_4284_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),zero_z3403309356797280102nteger)) ).
% not_numeral_le_zero
tff(fact_4285_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,numera5444537566228673987atural(N)),zero_z2226904508553997617atural)) ).
% not_numeral_le_zero
tff(fact_4286_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).
% not_numeral_le_zero
tff(fact_4287_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).
% not_numeral_le_zero
tff(fact_4288_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).
% not_numeral_le_zero
tff(fact_4289_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).
% mod_Suc_le_divisor
tff(fact_4290_zero__less__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).
% zero_less_numeral
tff(fact_4291_zero__less__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).
% zero_less_numeral
tff(fact_4292_zero__less__numeral,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% zero_less_numeral
tff(fact_4293_zero__less__numeral,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).
% zero_less_numeral
tff(fact_4294_zero__less__numeral,axiom,
! [N: num] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,zero_z2226904508553997617atural),numera5444537566228673987atural(N))) ).
% zero_less_numeral
tff(fact_4295_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).
% not_numeral_less_zero
tff(fact_4296_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).
% not_numeral_less_zero
tff(fact_4297_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),zero_z3403309356797280102nteger)) ).
% not_numeral_less_zero
tff(fact_4298_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).
% not_numeral_less_zero
tff(fact_4299_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,numera5444537566228673987atural(N)),zero_z2226904508553997617atural)) ).
% not_numeral_less_zero
tff(fact_4300_one__le__numeral,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% one_le_numeral
tff(fact_4301_one__le__numeral,axiom,
! [N: num] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,one_one_Code_natural),numera5444537566228673987atural(N))) ).
% one_le_numeral
tff(fact_4302_one__le__numeral,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).
% one_le_numeral
tff(fact_4303_one__le__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).
% one_le_numeral
tff(fact_4304_one__le__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).
% one_le_numeral
tff(fact_4305_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% nat_power_less_imp_less
tff(fact_4306_mod__eq__0D,axiom,
! [M: nat,D: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
=> ? [Q4: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q4) ) ).
% mod_eq_0D
tff(fact_4307_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M)) ) ).
% zmod_le_nonneg_dividend
tff(fact_4308_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L)) ) ).
% Euclidean_Division.pos_mod_bound
tff(fact_4309_neg__mod__bound,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).
% neg_mod_bound
tff(fact_4310_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).
% not_numeral_less_one
tff(fact_4311_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).
% not_numeral_less_one
tff(fact_4312_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)),one_one_Code_integer)) ).
% not_numeral_less_one
tff(fact_4313_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).
% not_numeral_less_one
tff(fact_4314_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,numera5444537566228673987atural(N)),one_one_Code_natural)) ).
% not_numeral_less_one
tff(fact_4315_le__mod__geq,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).
% le_mod_geq
tff(fact_4316_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).
% not_numeral_le_neg_numeral
tff(fact_4317_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).
% not_numeral_le_neg_numeral
tff(fact_4318_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_numeral_le_neg_numeral
tff(fact_4319_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% neg_numeral_le_numeral
tff(fact_4320_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).
% neg_numeral_le_numeral
tff(fact_4321_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).
% neg_numeral_le_numeral
tff(fact_4322_zero__neq__neg__numeral,axiom,
! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% zero_neq_neg_numeral
tff(fact_4323_zero__neq__neg__numeral,axiom,
! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).
% zero_neq_neg_numeral
tff(fact_4324_zero__neq__neg__numeral,axiom,
! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).
% zero_neq_neg_numeral
tff(fact_4325_one__plus__numeral__commute,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).
% one_plus_numeral_commute
tff(fact_4326_one__plus__numeral__commute,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).
% one_plus_numeral_commute
tff(fact_4327_one__plus__numeral__commute,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),one_one_Code_integer) ).
% one_plus_numeral_commute
tff(fact_4328_one__plus__numeral__commute,axiom,
! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).
% one_plus_numeral_commute
tff(fact_4329_one__plus__numeral__commute,axiom,
! [X: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),numera5444537566228673987atural(X)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(X)),one_one_Code_natural) ).
% one_plus_numeral_commute
tff(fact_4330_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
=> ? [Q4: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).
% zmod_eq_0D
tff(fact_4331_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
<=> ? [Q6: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q6) ) ).
% zmod_eq_0_iff
tff(fact_4332_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).
% zmod_zminus2_not_zero
tff(fact_4333_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).
% zmod_zminus1_not_zero
tff(fact_4334_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).
% neg_numeral_less_numeral
tff(fact_4335_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% neg_numeral_less_numeral
tff(fact_4336_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).
% neg_numeral_less_numeral
tff(fact_4337_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_numeral_less_neg_numeral
tff(fact_4338_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).
% not_numeral_less_neg_numeral
tff(fact_4339_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).
% not_numeral_less_neg_numeral
tff(fact_4340_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
<=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).
% nat_mod_eq_iff
tff(fact_4341_numeral__neq__neg__one,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).
% numeral_neq_neg_one
tff(fact_4342_numeral__neq__neg__one,axiom,
! [N: num] : aa_num_Code_integer(numera6620942414471956472nteger,N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% numeral_neq_neg_one
tff(fact_4343_numeral__neq__neg__one,axiom,
! [N: num] : aa_num_rat(numeral_numeral_rat,N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% numeral_neq_neg_one
tff(fact_4344_one__neq__neg__numeral,axiom,
! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% one_neq_neg_numeral
tff(fact_4345_one__neq__neg__numeral,axiom,
! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)) ).
% one_neq_neg_numeral
tff(fact_4346_one__neq__neg__numeral,axiom,
! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).
% one_neq_neg_numeral
tff(fact_4347_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4348_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4349_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4350_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4351_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4352_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4353_div__mult1__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).
% div_mult1_eq
tff(fact_4354_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_4355_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,B),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B),C))),C)) ).
% div_mult1_eq
tff(fact_4356_div__mult1__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,B),C))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,B),C))),C)) ).
% div_mult1_eq
tff(fact_4357_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_4358_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_4359_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_4360_cancel__div__mod__rules_I2_J,axiom,
! [B: code_natural,A: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_4361_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_4362_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_4363_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_4364_cancel__div__mod__rules_I1_J,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_4365_mod__div__decomp,axiom,
! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).
% mod_div_decomp
tff(fact_4366_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_4367_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) ).
% mod_div_decomp
tff(fact_4368_mod__div__decomp,axiom,
! [A: code_natural,B: code_natural] : A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)) ).
% mod_div_decomp
tff(fact_4369_div__mult__mod__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).
% div_mult_mod_eq
tff(fact_4370_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_4371_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) = A ).
% div_mult_mod_eq
tff(fact_4372_div__mult__mod__eq,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)) = A ).
% div_mult_mod_eq
tff(fact_4373_mod__div__mult__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_4374_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_4375_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_4376_mod__div__mult__eq,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_4377_mod__mult__div__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).
% mod_mult_div_eq
tff(fact_4378_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_4379_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))) = A ).
% mod_mult_div_eq
tff(fact_4380_mod__mult__div__eq,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B))) = A ).
% mod_mult_div_eq
tff(fact_4381_mult__div__mod__eq,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).
% mult_div_mod_eq
tff(fact_4382_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_4383_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) = A ).
% mult_div_mod_eq
tff(fact_4384_mult__div__mod__eq,axiom,
! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),B))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),B)) = A ).
% mult_div_mod_eq
tff(fact_4385_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).
% mod_le_divisor
tff(fact_4386_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).
% fact_mod
tff(fact_4387_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).
% fact_mod
tff(fact_4388_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,semiri2447717529341329178atural(N)),semiri2447717529341329178atural(M)) = zero_z2226904508553997617atural ) ) ).
% fact_mod
tff(fact_4389_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).
% fact_mod
tff(fact_4390_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger)) ).
% neg_numeral_le_zero
tff(fact_4391_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).
% neg_numeral_le_zero
tff(fact_4392_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).
% neg_numeral_le_zero
tff(fact_4393_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).
% not_zero_le_neg_numeral
tff(fact_4394_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).
% not_zero_le_neg_numeral
tff(fact_4395_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_zero_le_neg_numeral
tff(fact_4396_power__gt__expt,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).
% power_gt_expt
tff(fact_4397_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_zero_less_neg_numeral
tff(fact_4398_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N)))) ).
% not_zero_less_neg_numeral
tff(fact_4399_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).
% not_zero_less_neg_numeral
tff(fact_4400_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).
% neg_numeral_less_zero
tff(fact_4401_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))),zero_z3403309356797280102nteger)) ).
% neg_numeral_less_zero
tff(fact_4402_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).
% neg_numeral_less_zero
tff(fact_4403_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).
% nat_one_le_power
tff(fact_4404_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).
% not_one_le_neg_numeral
tff(fact_4405_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).
% not_one_le_neg_numeral
tff(fact_4406_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).
% not_one_le_neg_numeral
tff(fact_4407_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% not_numeral_le_neg_one
tff(fact_4408_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% not_numeral_le_neg_one
tff(fact_4409_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).
% not_numeral_le_neg_one
tff(fact_4410_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% neg_numeral_le_neg_one
tff(fact_4411_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% neg_numeral_le_neg_one
tff(fact_4412_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).
% neg_numeral_le_neg_one
tff(fact_4413_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M))) ).
% neg_one_le_numeral
tff(fact_4414_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).
% neg_one_le_numeral
tff(fact_4415_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).
% neg_one_le_numeral
tff(fact_4416_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer)) ).
% neg_numeral_le_one
tff(fact_4417_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).
% neg_numeral_le_one
tff(fact_4418_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).
% neg_numeral_le_one
tff(fact_4419_neg__mod__sign,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int)) ) ).
% neg_mod_sign
tff(fact_4420_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).
% Euclidean_Division.pos_mod_sign
tff(fact_4421_neg__mod__conj,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ) ).
% neg_mod_conj
tff(fact_4422_pos__mod__conj,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ) ).
% pos_mod_conj
tff(fact_4423_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
<=> ( ( K = zero_zero_int )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).
% zmod_trivial_iff
tff(fact_4424_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) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C) = B ) )
& ( ( C = zero_zero_rat )
=> ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_4425_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) )
<=> ( ( ( C != zero_zero_rat )
=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C) ) )
& ( ( C = zero_zero_rat )
=> ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_4426_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> ? [Q4: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ) ) ).
% nat_mod_eq_lemma
tff(fact_4427_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ~ ! [S2: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).
% mod_eq_nat2E
tff(fact_4428_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).
% mod_eq_nat1E
tff(fact_4429_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).
% neg_numeral_less_one
tff(fact_4430_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer)) ).
% neg_numeral_less_one
tff(fact_4431_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).
% neg_numeral_less_one
tff(fact_4432_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).
% neg_one_less_numeral
tff(fact_4433_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M))) ).
% neg_one_less_numeral
tff(fact_4434_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).
% neg_one_less_numeral
tff(fact_4435_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).
% not_numeral_less_neg_one
tff(fact_4436_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% not_numeral_less_neg_one
tff(fact_4437_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).
% not_numeral_less_neg_one
tff(fact_4438_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).
% not_one_less_neg_numeral
tff(fact_4439_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).
% not_one_less_neg_numeral
tff(fact_4440_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).
% not_one_less_neg_numeral
tff(fact_4441_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_4442_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_4443_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_4444_div__less__mono,axiom,
! [A3: nat,B3: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N) = zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N) = zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N))) ) ) ) ) ).
% div_less_mono
tff(fact_4445_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L))) ) ).
% abs_mod_less
tff(fact_4446_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ) ).
% zmod_zminus1_eq_if
tff(fact_4447_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ) ).
% zmod_zminus2_eq_if
tff(fact_4448_zdiv__mono__strict,axiom,
! [A3: int,B3: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N) = zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N) = zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N))) ) ) ) ) ).
% zdiv_mono_strict
tff(fact_4449_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).
% mod_mult2_eq
tff(fact_4450_div__mod__decomp,axiom,
! [A3: nat,N: 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),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N)) ).
% div_mod_decomp
tff(fact_4451_fact__le__power,axiom,
! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_4452_fact__le__power,axiom,
! [N: nat] : pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,semiri2447717529341329178atural(N)),aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_4453_fact__le__power,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_4454_fact__le__power,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_4455_fact__le__power,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_4456_div__mod__decomp__int,axiom,
! [A3: int,N: 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(aa_int_fun_int_int(divide_divide_int,A3),N)),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N)) ).
% div_mod_decomp_int
tff(fact_4457_fact__div__fact__le__pow,axiom,
! [R: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).
% fact_div_fact_le_pow
tff(fact_4458_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).
% mod_mult2_eq'
tff(fact_4459_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).
% mod_mult2_eq'
tff(fact_4460_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).
% mod_mult2_eq'
tff(fact_4461_mod__mult2__eq_H,axiom,
! [A: code_natural,M: nat,N: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,A),aa_nat_Code_natural(semiri3763490453095760265atural,M))),aa_nat_Code_natural(semiri3763490453095760265atural,N)))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,A),aa_nat_Code_natural(semiri3763490453095760265atural,M))) ).
% mod_mult2_eq'
tff(fact_4462_split__mod,axiom,
! [P: fun_nat_bool,M: nat,N: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
<=> ( ( ( N = zero_zero_nat )
=> pp(aa_nat_bool(P,M)) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
=> ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J3) )
=> pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).
% split_mod
tff(fact_4463_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_4464_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_4465_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% field_char_0_class.of_nat_div
tff(fact_4466_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) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B ) )
& ( ( C = zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_4467_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)) )
<=> ( ( ( C != zero_zero_rat )
=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) ) )
& ( ( C = zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_4468_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).
% mod_pos_neg_trivial
tff(fact_4469_mod__pos__geq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).
% mod_pos_geq
tff(fact_4470_array__of__list__make,axiom,
! [Xs: list_nat] : array_of_list_nat(Xs) = array_make_nat(aa_list_nat_nat(size_size_list_nat,Xs),nth_nat(Xs)) ).
% array_of_list_make
tff(fact_4471_array__of__list__make,axiom,
! [Xs: list_int] : array_of_list_int(Xs) = array_make_int(aa_list_int_nat(size_size_list_int,Xs),nth_int(Xs)) ).
% array_of_list_make
tff(fact_4472_array__of__list__make,axiom,
! [Xs: list_bool] : array_of_list_bool(Xs) = array_make_bool(aa_list_bool_nat(size_size_list_bool,Xs),nth_bool(Xs)) ).
% array_of_list_make
tff(fact_4473_nat__power__eq,axiom,
! [Z: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).
% nat_power_eq
tff(fact_4474_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4475_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4476_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4477_fold__atLeastAtMost__nat_Osimps,axiom,
! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = Acc ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = 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),Acc)) ) ) ) ).
% fold_atLeastAtMost_nat.simps
tff(fact_4478_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
=> ( Y = Xc ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
tff(fact_4479_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_4480_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_4481_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).
% Suc_times_mod_eq
tff(fact_4482_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_4483_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_4484_int__mod__pos__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).
% int_mod_pos_eq
tff(fact_4485_int__mod__neg__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).
% int_mod_neg_eq
tff(fact_4486_split__zmod,axiom,
! [P: fun_int_bool,N: int,K: int] :
( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
<=> ( ( ( K = zero_zero_int )
=> pp(aa_int_bool(P,N)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ! [I3: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
=> pp(aa_int_bool(P,J3)) ) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ! [I3: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
=> pp(aa_int_bool(P,J3)) ) ) ) ) ).
% split_zmod
tff(fact_4487_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).
% minus_mod_int_eq
tff(fact_4488_zmod__minus1,axiom,
! [B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).
% zmod_minus1
tff(fact_4489_zmod__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).
% zmod_zmult2_eq
tff(fact_4490_zdiv__zminus1__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).
% zdiv_zminus1_eq_if
tff(fact_4491_zdiv__zminus2__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).
% zdiv_zminus2_eq_if
tff(fact_4492_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_4493_sorted__in__between,axiom,
! [I: nat,J: nat,L: list_bool,X: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,L)))
=> ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L))
=> ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(L),I)),X))
=> ( ord_less_bool(X,aa_nat_bool(nth_bool(L),J))
=> ~ ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),J))
=> ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(L),K3)),X))
=> ~ ord_less_bool(X,aa_nat_bool(nth_bool(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat))) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4494_sorted__in__between,axiom,
! [I: nat,J: nat,L: list_Code_integer,X: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s3445333598471063425nteger(L)))
=> ( sorted710888440204495920nteger(ord_le3102999989581377725nteger,L)
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(nth_Code_integer(L),I)),X))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),aa_nat_Code_integer(nth_Code_integer(L),J)))
=> ~ ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),J))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(nth_Code_integer(L),K3)),X))
=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),aa_nat_Code_integer(nth_Code_integer(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4495_sorted__in__between,axiom,
! [I: nat,J: nat,L: list_rat,X: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_rat(L)))
=> ( sorted_wrt_rat(ord_less_eq_rat,L)
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(nth_rat(L),J)))
=> ~ ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),J))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(L),K3)),X))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(nth_rat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4496_sorted__in__between,axiom,
! [I: nat,J: nat,L: list_num,X: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
=> ( sorted_wrt_num(ord_less_eq_num,L)
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),I)),X))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),aa_nat_num(nth_num(L),J)))
=> ~ ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),J))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),K3)),X))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),aa_nat_num(nth_num(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4497_sorted__in__between,axiom,
! [I: nat,J: nat,L: list_nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
=> ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(nth_nat(L),J)))
=> ~ ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),K3)),X))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4498_sorted__in__between,axiom,
! [I: nat,J: nat,L: list_int,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,L)))
=> ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),I)),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(nth_int(L),J)))
=> ~ ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),K3)),X))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4499_diff__numeral__special_I6_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,M)) ).
% diff_numeral_special(6)
tff(fact_4500_diff__numeral__special_I6_J,axiom,
! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M)) ).
% diff_numeral_special(6)
tff(fact_4501_diff__numeral__special_I6_J,axiom,
! [M: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M)) ).
% diff_numeral_special(6)
tff(fact_4502_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).
% diff_numeral_special(5)
tff(fact_4503_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) ).
% diff_numeral_special(5)
tff(fact_4504_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N))) ).
% diff_numeral_special(5)
tff(fact_4505_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).
% add_neg_numeral_special(5)
tff(fact_4506_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N))) ).
% add_neg_numeral_special(5)
tff(fact_4507_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_rat_rat(aa_rat_fun_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,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N))) ).
% add_neg_numeral_special(5)
tff(fact_4508_add__neg__numeral__special_I6_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,M))) ).
% add_neg_numeral_special(6)
tff(fact_4509_add__neg__numeral__special_I6_J,axiom,
! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M))) ).
% add_neg_numeral_special(6)
tff(fact_4510_add__neg__numeral__special_I6_J,axiom,
! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),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,M))) ).
% add_neg_numeral_special(6)
tff(fact_4511_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
=> ( aa_nat_nat(nth_nat(rotate_nat(M,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).
% nth_rotate
tff(fact_4512_nth__rotate,axiom,
! [N: nat,Xs: list_int,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
=> ( aa_nat_int(nth_int(rotate_int(M,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_list_int_nat(size_size_list_int,Xs))) ) ) ).
% nth_rotate
tff(fact_4513_nth__rotate,axiom,
! [N: nat,Xs: list_bool,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
=> ( pp(aa_nat_bool(nth_bool(rotate_bool(M,Xs)),N))
<=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_list_bool_nat(size_size_list_bool,Xs)))) ) ) ).
% nth_rotate
tff(fact_4514_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% nth_Cons_pos
tff(fact_4515_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% nth_Cons_pos
tff(fact_4516_nth__Cons__pos,axiom,
! [N: nat,X: char,Xs: list_char] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( nth_char(cons_char(X,Xs),N) = nth_char(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% nth_Cons_pos
tff(fact_4517_length__rotate,axiom,
! [N: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,rotate_nat(N,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).
% length_rotate
tff(fact_4518_length__rotate,axiom,
! [N: nat,Xs: list_int] : aa_list_int_nat(size_size_list_int,rotate_int(N,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).
% length_rotate
tff(fact_4519_length__rotate,axiom,
! [N: nat,Xs: list_bool] : aa_list_bool_nat(size_size_list_bool,rotate_bool(N,Xs)) = aa_list_bool_nat(size_size_list_bool,Xs) ).
% length_rotate
tff(fact_4520_nth__Cons__0,axiom,
! [X: int,Xs: list_int] : aa_nat_int(nth_int(cons_int(X,Xs)),zero_zero_nat) = X ).
% nth_Cons_0
tff(fact_4521_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),zero_zero_nat) = X ).
% nth_Cons_0
tff(fact_4522_nth__Cons__0,axiom,
! [X: char,Xs: list_char] : nth_char(cons_char(X,Xs),zero_zero_nat) = X ).
% nth_Cons_0
tff(fact_4523_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
=> ( rotate_nat(N,Xs) = Xs ) ) ).
% rotate_length01
tff(fact_4524_rotate__length01,axiom,
! [Xs: list_int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat))
=> ( rotate_int(N,Xs) = Xs ) ) ).
% rotate_length01
tff(fact_4525_rotate__length01,axiom,
! [Xs: list_bool,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat))
=> ( rotate_bool(N,Xs) = Xs ) ) ).
% rotate_length01
tff(fact_4526_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)) = zero_zero_nat )
=> ( rotate_nat(N,Xs) = Xs ) ) ).
% rotate_id
tff(fact_4527_rotate__id,axiom,
! [N: nat,Xs: list_int] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_int_nat(size_size_list_int,Xs)) = zero_zero_nat )
=> ( rotate_int(N,Xs) = Xs ) ) ).
% rotate_id
tff(fact_4528_rotate__id,axiom,
! [N: nat,Xs: list_bool] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)) = zero_zero_nat )
=> ( rotate_bool(N,Xs) = Xs ) ) ).
% rotate_id
tff(fact_4529_nth__Cons__numeral,axiom,
! [X: int,Xs: list_int,V: num] : aa_nat_int(nth_int(cons_int(X,Xs)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).
% nth_Cons_numeral
tff(fact_4530_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).
% nth_Cons_numeral
tff(fact_4531_nth__Cons__numeral,axiom,
! [X: char,Xs: list_char,V: num] : nth_char(cons_char(X,Xs),aa_num_nat(numeral_numeral_nat,V)) = nth_char(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).
% nth_Cons_numeral
tff(fact_4532_mult__inc,axiom,
! [X: num,Y: num] : aa_num_num(times_times_num(X),aa_num_num(inc,Y)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(X),Y)),X) ).
% mult_inc
tff(fact_4533_sorted2,axiom,
! [X: rat,Y: rat,Zs: list_rat] :
( sorted_wrt_rat(ord_less_eq_rat,cons_rat(X,cons_rat(Y,Zs)))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
& sorted_wrt_rat(ord_less_eq_rat,cons_rat(Y,Zs)) ) ) ).
% sorted2
tff(fact_4534_sorted2,axiom,
! [X: num,Y: num,Zs: list_num] :
( sorted_wrt_num(ord_less_eq_num,cons_num(X,cons_num(Y,Zs)))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
& sorted_wrt_num(ord_less_eq_num,cons_num(Y,Zs)) ) ) ).
% sorted2
tff(fact_4535_sorted2,axiom,
! [X: nat,Y: nat,Zs: list_nat] :
( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),cons_nat(X,cons_nat(Y,Zs))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
& pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),cons_nat(Y,Zs))) ) ) ).
% sorted2
tff(fact_4536_sorted2,axiom,
! [X: int,Y: int,Zs: list_int] :
( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),cons_int(X,cons_int(Y,Zs))))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
& pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),cons_int(Y,Zs))) ) ) ).
% sorted2
tff(fact_4537_add__inc,axiom,
! [X: num,Y: num] : aa_num_num(plus_plus_num(X),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(plus_plus_num(X),Y)) ).
% add_inc
tff(fact_4538_list__tail__coinc,axiom,
! [N1: int,R1: list_int,N22: int,R22: list_int] :
( ( cons_int(N1,R1) = cons_int(N22,R22) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
tff(fact_4539_list__tail__coinc,axiom,
! [N1: nat,R1: list_nat,N22: nat,R22: list_nat] :
( ( cons_nat(N1,R1) = cons_nat(N22,R22) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
tff(fact_4540_list__tail__coinc,axiom,
! [N1: char,R1: list_char,N22: char,R22: list_char] :
( ( cons_char(N1,R1) = cons_char(N22,R22) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
tff(fact_4541_sorted__wrt__true,axiom,
! [Xs: list_nat] : pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_cb),Xs)) ).
% sorted_wrt_true
tff(fact_4542_sorted__wrt__true,axiom,
! [Xs: list_int] : pp(aa_list_int_bool(sorted_wrt_int(aTP_Lamm_cc),Xs)) ).
% sorted_wrt_true
tff(fact_4543_sorted1,axiom,
! [X: rat] : sorted_wrt_rat(ord_less_eq_rat,cons_rat(X,nil_rat)) ).
% sorted1
tff(fact_4544_sorted1,axiom,
! [X: num] : sorted_wrt_num(ord_less_eq_num,cons_num(X,nil_num)) ).
% sorted1
tff(fact_4545_sorted1,axiom,
! [X: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),cons_nat(X,nil_nat))) ).
% sorted1
tff(fact_4546_sorted1,axiom,
! [X: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),cons_int(X,nil_int))) ).
% sorted1
tff(fact_4547_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li582272206764200913t_bool,R2: fun_int_fun_int_bool,Xs: list_int,Ys: list_int] :
( ! [Xs2: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs2),nil_int))
=> ( ! [Ys3: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),Ys3))
=> ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(R2,X2),Y2))
=> ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs2),cons_int(Y2,Ys3)))
=> pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),cons_int(Y2,Ys3))) ) )
=> ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(R2,X2),Y2))
=> ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),Ys3))
=> pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),cons_int(Y2,Ys3))) ) )
=> pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4548_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li1679355187408199469t_bool,R2: fun_nat_fun_int_bool,Xs: list_nat,Ys: list_int] :
( ! [Xs2: list_nat] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs2),nil_int))
=> ( ! [Ys3: list_int] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),Ys3))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int] :
( pp(aa_int_bool(aa_nat_fun_int_bool(R2,X2),Y2))
=> ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs2),cons_int(Y2,Ys3)))
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3))) ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int] :
( ~ pp(aa_int_bool(aa_nat_fun_int_bool(R2,X2),Y2))
=> ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),Ys3))
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3))) ) )
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4549_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li3246207939727701100t_bool,R2: fun_ch2282331315503596684t_bool,Xs: list_char,Ys: list_int] :
( ! [Xs2: list_char] : pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,Xs2),nil_int))
=> ( ! [Ys3: list_int] : pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,nil_char),Ys3))
=> ( ! [X2: char,Xs2: list_char,Y2: int,Ys3: list_int] :
( pp(aa_int_bool(aa_char_fun_int_bool(R2,X2),Y2))
=> ( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,Xs2),cons_int(Y2,Ys3)))
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,cons_char(X2,Xs2)),cons_int(Y2,Ys3))) ) )
=> ( ! [X2: char,Xs2: list_char,Y2: int,Ys3: list_int] :
( ~ pp(aa_int_bool(aa_char_fun_int_bool(R2,X2),Y2))
=> ( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,cons_char(X2,Xs2)),Ys3))
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,cons_char(X2,Xs2)),cons_int(Y2,Ys3))) ) )
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4550_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li7501238334638356269t_bool,R2: fun_int_fun_nat_bool,Xs: list_int,Ys: list_nat] :
( ! [Xs2: list_int] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs2),nil_nat))
=> ( ! [Ys3: list_nat] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),Ys3))
=> ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs2),cons_nat(Y2,Ys3)))
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),cons_nat(Y2,Ys3))) ) )
=> ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat] :
( ~ pp(aa_nat_bool(aa_int_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),Ys3))
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),cons_nat(Y2,Ys3))) ) )
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4551_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li8598321315282354825t_bool,R2: fun_nat_fun_nat_bool,Xs: list_nat,Ys: list_nat] :
( ! [Xs2: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs2),nil_nat))
=> ( ! [Ys3: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),Ys3))
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs2),cons_nat(Y2,Ys3)))
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys3))) ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),Ys3))
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys3))) ) )
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4552_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li941802030747080648t_bool,R2: fun_ch2742614267150243304t_bool,Xs: list_char,Ys: list_nat] :
( ! [Xs2: list_char] : pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,Xs2),nil_nat))
=> ( ! [Ys3: list_nat] : pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,nil_char),Ys3))
=> ( ! [X2: char,Xs2: list_char,Y2: nat,Ys3: list_nat] :
( pp(aa_nat_bool(aa_char_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,Xs2),cons_nat(Y2,Ys3)))
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,cons_char(X2,Xs2)),cons_nat(Y2,Ys3))) ) )
=> ( ! [X2: char,Xs2: list_char,Y2: nat,Ys3: list_nat] :
( ~ pp(aa_nat_bool(aa_char_fun_nat_bool(R2,X2),Y2))
=> ( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,cons_char(X2,Xs2)),Ys3))
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,cons_char(X2,Xs2)),cons_nat(Y2,Ys3))) ) )
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4553_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li7776623116333600876r_bool,R2: fun_in892379244202657548r_bool,Xs: list_int,Ys: list_char] :
( ! [Xs2: list_int] : pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,Xs2),nil_char))
=> ( ! [Ys3: list_char] : pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,nil_int),Ys3))
=> ( ! [X2: int,Xs2: list_int,Y2: char,Ys3: list_char] :
( pp(aa_char_bool(aa_int_fun_char_bool(R2,X2),Y2))
=> ( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,Xs2),cons_char(Y2,Ys3)))
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,cons_int(X2,Xs2)),cons_char(Y2,Ys3))) ) )
=> ( ! [X2: int,Xs2: list_int,Y2: char,Ys3: list_char] :
( ~ pp(aa_char_bool(aa_int_fun_char_bool(R2,X2),Y2))
=> ( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,cons_int(X2,Xs2)),Ys3))
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,cons_int(X2,Xs2)),cons_char(Y2,Ys3))) ) )
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4554_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li4510401072603889608r_bool,R2: fun_na6281512180173406824r_bool,Xs: list_nat,Ys: list_char] :
( ! [Xs2: list_nat] : pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,Xs2),nil_char))
=> ( ! [Ys3: list_char] : pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,nil_nat),Ys3))
=> ( ! [X2: nat,Xs2: list_nat,Y2: char,Ys3: list_char] :
( pp(aa_char_bool(aa_nat_fun_char_bool(R2,X2),Y2))
=> ( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,Xs2),cons_char(Y2,Ys3)))
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,cons_nat(X2,Xs2)),cons_char(Y2,Ys3))) ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: char,Ys3: list_char] :
( ~ pp(aa_char_bool(aa_nat_fun_char_bool(R2,X2),Y2))
=> ( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,cons_nat(X2,Xs2)),Ys3))
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,cons_nat(X2,Xs2)),cons_char(Y2,Ys3))) ) )
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4555_mergesort__by__rel__merge__induct,axiom,
! [P: fun_li8666427833891403271r_bool,R2: fun_ch9086500746998582951r_bool,Xs: list_char,Ys: list_char] :
( ! [Xs2: list_char] : pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,Xs2),nil_char))
=> ( ! [Ys3: list_char] : pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,nil_char),Ys3))
=> ( ! [X2: char,Xs2: list_char,Y2: char,Ys3: list_char] :
( pp(aa_char_bool(aa_cha4216456086387821712r_bool(R2,X2),Y2))
=> ( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,Xs2),cons_char(Y2,Ys3)))
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,cons_char(X2,Xs2)),cons_char(Y2,Ys3))) ) )
=> ( ! [X2: char,Xs2: list_char,Y2: char,Ys3: list_char] :
( ~ pp(aa_char_bool(aa_cha4216456086387821712r_bool(R2,X2),Y2))
=> ( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,cons_char(X2,Xs2)),Ys3))
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,cons_char(X2,Xs2)),cons_char(Y2,Ys3))) ) )
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4556_list__induct__first2,axiom,
! [P: fun_list_int_bool,Xs: list_int] :
( pp(aa_list_int_bool(P,nil_int))
=> ( ! [X2: int] : pp(aa_list_int_bool(P,cons_int(X2,nil_int)))
=> ( ! [X1: int,X23: int,Xs2: list_int] :
( pp(aa_list_int_bool(P,Xs2))
=> pp(aa_list_int_bool(P,cons_int(X1,cons_int(X23,Xs2)))) )
=> pp(aa_list_int_bool(P,Xs)) ) ) ) ).
% list_induct_first2
tff(fact_4557_list__induct__first2,axiom,
! [P: fun_list_nat_bool,Xs: list_nat] :
( pp(aa_list_nat_bool(P,nil_nat))
=> ( ! [X2: nat] : pp(aa_list_nat_bool(P,cons_nat(X2,nil_nat)))
=> ( ! [X1: nat,X23: nat,Xs2: list_nat] :
( pp(aa_list_nat_bool(P,Xs2))
=> pp(aa_list_nat_bool(P,cons_nat(X1,cons_nat(X23,Xs2)))) )
=> pp(aa_list_nat_bool(P,Xs)) ) ) ) ).
% list_induct_first2
tff(fact_4558_list__induct__first2,axiom,
! [P: fun_list_char_bool,Xs: list_char] :
( pp(aa_list_char_bool(P,nil_char))
=> ( ! [X2: char] : pp(aa_list_char_bool(P,cons_char(X2,nil_char)))
=> ( ! [X1: char,X23: char,Xs2: list_char] :
( pp(aa_list_char_bool(P,Xs2))
=> pp(aa_list_char_bool(P,cons_char(X1,cons_char(X23,Xs2)))) )
=> pp(aa_list_char_bool(P,Xs)) ) ) ) ).
% list_induct_first2
tff(fact_4559_list__2pre__induct,axiom,
! [P: fun_li582272206764200913t_bool,W1: list_int,W22: list_int] :
( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),nil_int))
=> ( ! [E3: int,W12: list_int,W23: list_int] :
( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W12),W23))
=> pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(E3,W12)),W23)) )
=> ( ! [E3: int,W13: list_int,W24: list_int] :
( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W13),W24))
=> pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W13),cons_int(E3,W24))) )
=> pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4560_list__2pre__induct,axiom,
! [P: fun_li7501238334638356269t_bool,W1: list_int,W22: list_nat] :
( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),nil_nat))
=> ( ! [E3: int,W12: list_int,W23: list_nat] :
( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W12),W23))
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(E3,W12)),W23)) )
=> ( ! [E3: nat,W13: list_int,W24: list_nat] :
( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W13),W24))
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W13),cons_nat(E3,W24))) )
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4561_list__2pre__induct,axiom,
! [P: fun_li7776623116333600876r_bool,W1: list_int,W22: list_char] :
( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,nil_int),nil_char))
=> ( ! [E3: int,W12: list_int,W23: list_char] :
( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,W12),W23))
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,cons_int(E3,W12)),W23)) )
=> ( ! [E3: char,W13: list_int,W24: list_char] :
( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,W13),W24))
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,W13),cons_char(E3,W24))) )
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4562_list__2pre__induct,axiom,
! [P: fun_li1679355187408199469t_bool,W1: list_nat,W22: list_int] :
( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),nil_int))
=> ( ! [E3: nat,W12: list_nat,W23: list_int] :
( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W12),W23))
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(E3,W12)),W23)) )
=> ( ! [E3: int,W13: list_nat,W24: list_int] :
( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W13),W24))
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W13),cons_int(E3,W24))) )
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4563_list__2pre__induct,axiom,
! [P: fun_li8598321315282354825t_bool,W1: list_nat,W22: list_nat] :
( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),nil_nat))
=> ( ! [E3: nat,W12: list_nat,W23: list_nat] :
( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W12),W23))
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(E3,W12)),W23)) )
=> ( ! [E3: nat,W13: list_nat,W24: list_nat] :
( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W13),W24))
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W13),cons_nat(E3,W24))) )
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4564_list__2pre__induct,axiom,
! [P: fun_li4510401072603889608r_bool,W1: list_nat,W22: list_char] :
( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,nil_nat),nil_char))
=> ( ! [E3: nat,W12: list_nat,W23: list_char] :
( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,W12),W23))
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,cons_nat(E3,W12)),W23)) )
=> ( ! [E3: char,W13: list_nat,W24: list_char] :
( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,W13),W24))
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,W13),cons_char(E3,W24))) )
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4565_list__2pre__induct,axiom,
! [P: fun_li3246207939727701100t_bool,W1: list_char,W22: list_int] :
( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,nil_char),nil_int))
=> ( ! [E3: char,W12: list_char,W23: list_int] :
( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,W12),W23))
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,cons_char(E3,W12)),W23)) )
=> ( ! [E3: int,W13: list_char,W24: list_int] :
( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,W13),W24))
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,W13),cons_int(E3,W24))) )
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4566_list__2pre__induct,axiom,
! [P: fun_li941802030747080648t_bool,W1: list_char,W22: list_nat] :
( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,nil_char),nil_nat))
=> ( ! [E3: char,W12: list_char,W23: list_nat] :
( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,W12),W23))
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,cons_char(E3,W12)),W23)) )
=> ( ! [E3: nat,W13: list_char,W24: list_nat] :
( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,W13),W24))
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,W13),cons_nat(E3,W24))) )
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4567_list__2pre__induct,axiom,
! [P: fun_li8666427833891403271r_bool,W1: list_char,W22: list_char] :
( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,nil_char),nil_char))
=> ( ! [E3: char,W12: list_char,W23: list_char] :
( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,W12),W23))
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,cons_char(E3,W12)),W23)) )
=> ( ! [E3: char,W13: list_char,W24: list_char] :
( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,W13),W24))
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,W13),cons_char(E3,W24))) )
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(P,W1),W22)) ) ) ) ).
% list_2pre_induct
tff(fact_4568_neq__NilE,axiom,
! [L: list_int] :
( ( L != nil_int )
=> ~ ! [X2: int,Xs2: list_int] : L != cons_int(X2,Xs2) ) ).
% neq_NilE
tff(fact_4569_neq__NilE,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ~ ! [X2: nat,Xs2: list_nat] : L != cons_nat(X2,Xs2) ) ).
% neq_NilE
tff(fact_4570_neq__NilE,axiom,
! [L: list_char] :
( ( L != nil_char )
=> ~ ! [X2: char,Xs2: list_char] : L != cons_char(X2,Xs2) ) ).
% neq_NilE
tff(fact_4571_strict__sorted__imp__sorted,axiom,
! [Xs: list_Code_integer] :
( sorted710888440204495920nteger(ord_le6747313008572928689nteger,Xs)
=> sorted710888440204495920nteger(ord_le3102999989581377725nteger,Xs) ) ).
% strict_sorted_imp_sorted
tff(fact_4572_strict__sorted__imp__sorted,axiom,
! [Xs: list_rat] :
( sorted_wrt_rat(ord_less_rat,Xs)
=> sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).
% strict_sorted_imp_sorted
tff(fact_4573_strict__sorted__imp__sorted,axiom,
! [Xs: list_num] :
( sorted_wrt_num(ord_less_num,Xs)
=> sorted_wrt_num(ord_less_eq_num,Xs) ) ).
% strict_sorted_imp_sorted
tff(fact_4574_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Xs))
=> pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs)) ) ).
% strict_sorted_imp_sorted
tff(fact_4575_strict__sorted__imp__sorted,axiom,
! [Xs: list_int] :
( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Xs))
=> pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs)) ) ).
% strict_sorted_imp_sorted
tff(fact_4576_sorted0,axiom,
sorted_wrt_rat(ord_less_eq_rat,nil_rat) ).
% sorted0
tff(fact_4577_sorted0,axiom,
sorted_wrt_num(ord_less_eq_num,nil_num) ).
% sorted0
tff(fact_4578_sorted0,axiom,
pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),nil_nat)) ).
% sorted0
tff(fact_4579_sorted0,axiom,
pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),nil_int)) ).
% sorted0
tff(fact_4580_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ns)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).
% sorted_wrt_less_idx
tff(fact_4581_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_char,Ws: list_char,P: fun_li313415298244683037r_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = size_size_list_char(Zs) )
=> ( ( size_size_list_char(Zs) = size_size_list_char(Ws) )
=> ( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(aa_lis4488968080555687309r_bool(aa_lis586665551069413254r_bool(P,nil_nat),nil_int),nil_char),nil_char))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: char,Zs2: list_char,W2: char,Ws2: list_char] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = size_size_list_char(Zs2) )
=> ( ( size_size_list_char(Zs2) = size_size_list_char(Ws2) )
=> ( pp(aa_list_char_bool(aa_lis69393864293561840r_bool(aa_lis4488968080555687309r_bool(aa_lis586665551069413254r_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(aa_lis4488968080555687309r_bool(aa_lis586665551069413254r_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_char(Z2,Zs2)),cons_char(W2,Ws2))) ) ) ) )
=> pp(aa_list_char_bool(aa_lis69393864293561840r_bool(aa_lis4488968080555687309r_bool(aa_lis586665551069413254r_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4582_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_char,Ws: list_nat,P: fun_li213583066599728606t_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = size_size_list_char(Zs) )
=> ( ( size_size_list_char(Zs) = aa_list_nat_nat(size_size_list_nat,Ws) )
=> ( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(aa_lis1332358589793051982t_bool(aa_lis5462500619714208967t_bool(P,nil_nat),nil_int),nil_char),nil_nat))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: char,Zs2: list_char,W2: nat,Ws2: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = size_size_list_char(Zs2) )
=> ( ( size_size_list_char(Zs2) = aa_list_nat_nat(size_size_list_nat,Ws2) )
=> ( pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(aa_lis1332358589793051982t_bool(aa_lis5462500619714208967t_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(aa_lis1332358589793051982t_bool(aa_lis5462500619714208967t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_char(Z2,Zs2)),cons_nat(W2,Ws2))) ) ) ) )
=> pp(aa_list_nat_bool(aa_lis6166431010465545265t_bool(aa_lis1332358589793051982t_bool(aa_lis5462500619714208967t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4583_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_char,Ws: list_int,P: fun_li2609373428194598530t_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = size_size_list_char(Zs) )
=> ( ( size_size_list_char(Zs) = aa_list_int_nat(size_size_list_int,Ws) )
=> ( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(aa_lis3636764498773672434t_bool(aa_lis6149076652754759531t_bool(P,nil_nat),nil_int),nil_char),nil_int))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: char,Zs2: list_char,W2: int,Ws2: list_int] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = size_size_list_char(Zs2) )
=> ( ( size_size_list_char(Zs2) = aa_list_int_nat(size_size_list_int,Ws2) )
=> ( pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(aa_lis3636764498773672434t_bool(aa_lis6149076652754759531t_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(aa_lis3636764498773672434t_bool(aa_lis6149076652754759531t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_char(Z2,Zs2)),cons_int(W2,Ws2))) ) ) ) )
=> pp(aa_list_int_bool(aa_lis5706148058818898645t_bool(aa_lis3636764498773672434t_bool(aa_lis6149076652754759531t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4584_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_char,Ws: list_bool,P: fun_li748896040850507121l_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = size_size_list_char(Zs) )
=> ( ( size_size_list_char(Zs) = aa_list_bool_nat(size_size_list_bool,Ws) )
=> ( pp(aa_list_bool_bool(aa_lis1499569060329921180l_bool(aa_lis3183881253345691349l_bool(aa_lis4922630141027518642l_bool(P,nil_nat),nil_int),nil_char),nil_bool))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: char,Zs2: list_char,W2: bool,Ws2: list_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = size_size_list_char(Zs2) )
=> ( ( size_size_list_char(Zs2) = aa_list_bool_nat(size_size_list_bool,Ws2) )
=> ( pp(aa_list_bool_bool(aa_lis1499569060329921180l_bool(aa_lis3183881253345691349l_bool(aa_lis4922630141027518642l_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_bool_bool(aa_lis1499569060329921180l_bool(aa_lis3183881253345691349l_bool(aa_lis4922630141027518642l_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_char(Z2,Zs2)),cons_bool(W2,Ws2))) ) ) ) )
=> pp(aa_list_bool_bool(aa_lis1499569060329921180l_bool(aa_lis3183881253345691349l_bool(aa_lis4922630141027518642l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4585_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_nat,Ws: list_char,P: fun_li1197399218159603166r_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs) = size_size_list_char(Ws) )
=> ( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(aa_lis4900957631649860942r_bool(aa_lis241919026908056775r_bool(P,nil_nat),nil_int),nil_nat),nil_char))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: nat,Zs2: list_nat,W2: char,Ws2: list_char] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = aa_list_nat_nat(size_size_list_nat,Zs2) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs2) = size_size_list_char(Ws2) )
=> ( pp(aa_list_char_bool(aa_lis121307160345442225r_bool(aa_lis4900957631649860942r_bool(aa_lis241919026908056775r_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(aa_lis4900957631649860942r_bool(aa_lis241919026908056775r_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_nat(Z2,Zs2)),cons_char(W2,Ws2))) ) ) ) )
=> pp(aa_list_char_bool(aa_lis121307160345442225r_bool(aa_lis4900957631649860942r_bool(aa_lis241919026908056775r_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4586_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_nat,Ws: list_nat,P: fun_li2409054529390391199t_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs) = aa_list_nat_nat(size_size_list_nat,Ws) )
=> ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aa_lis8639271045199468047t_bool(aa_lis7035289734060725000t_bool(P,nil_nat),nil_int),nil_nat),nil_nat))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = aa_list_nat_nat(size_size_list_nat,Zs2) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs2) = aa_list_nat_nat(size_size_list_nat,Ws2) )
=> ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aa_lis8639271045199468047t_bool(aa_lis7035289734060725000t_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aa_lis8639271045199468047t_bool(aa_lis7035289734060725000t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_nat(Z2,Zs2)),cons_nat(W2,Ws2))) ) ) ) )
=> pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aa_lis8639271045199468047t_bool(aa_lis7035289734060725000t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4587_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_nat,Ws: list_int,P: fun_li4804844890985261123t_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs) = aa_list_int_nat(size_size_list_int,Ws) )
=> ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(aa_lis1720304917325312691t_bool(aa_lis7721865767101275564t_bool(P,nil_nat),nil_int),nil_nat),nil_int))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: nat,Zs2: list_nat,W2: int,Ws2: list_int] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = aa_list_nat_nat(size_size_list_nat,Zs2) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs2) = aa_list_int_nat(size_size_list_int,Ws2) )
=> ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(aa_lis1720304917325312691t_bool(aa_lis7721865767101275564t_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(aa_lis1720304917325312691t_bool(aa_lis7721865767101275564t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_nat(Z2,Zs2)),cons_int(W2,Ws2))) ) ) ) )
=> pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(aa_lis1720304917325312691t_bool(aa_lis7721865767101275564t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4588_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_nat,Ws: list_bool,P: fun_li7750400195056479216l_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs) = aa_list_bool_nat(size_size_list_bool,Ws) )
=> ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis9136032189843949140l_bool(aa_lis5676137900610579121l_bool(P,nil_nat),nil_int),nil_nat),nil_bool))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: nat,Zs2: list_nat,W2: bool,Ws2: list_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = aa_list_nat_nat(size_size_list_nat,Zs2) )
=> ( ( aa_list_nat_nat(size_size_list_nat,Zs2) = aa_list_bool_nat(size_size_list_bool,Ws2) )
=> ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis9136032189843949140l_bool(aa_lis5676137900610579121l_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis9136032189843949140l_bool(aa_lis5676137900610579121l_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_nat(Z2,Zs2)),cons_bool(W2,Ws2))) ) ) ) )
=> pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis9136032189843949140l_bool(aa_lis5676137900610579121l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4589_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_char,P: fun_li9008947015998503554r_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = aa_list_int_nat(size_size_list_int,Zs) )
=> ( ( aa_list_int_nat(size_size_list_int,Zs) = size_size_list_char(Ws) )
=> ( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(aa_lis8167179675379572210r_bool(aa_lis8293019882697296747r_bool(P,nil_nat),nil_int),nil_int),nil_char))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: int,Zs2: list_int,W2: char,Ws2: list_char] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = aa_list_int_nat(size_size_list_int,Zs2) )
=> ( ( aa_list_int_nat(size_size_list_int,Zs2) = size_size_list_char(Ws2) )
=> ( pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(aa_lis8167179675379572210r_bool(aa_lis8293019882697296747r_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(aa_lis8167179675379572210r_bool(aa_lis8293019882697296747r_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_int(Z2,Zs2)),cons_char(W2,Ws2))) ) ) ) )
=> pp(aa_list_char_bool(aa_lis3387529204075153493r_bool(aa_lis8167179675379572210r_bool(aa_lis8293019882697296747r_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4590_list__induct4,axiom,
! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_nat,P: fun_li5675276573120102467t_bool] :
( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys) = aa_list_int_nat(size_size_list_int,Zs) )
=> ( ( aa_list_int_nat(size_size_list_int,Zs) = aa_list_nat_nat(size_size_list_nat,Ws) )
=> ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7542188064555469491t_bool(aa_lis8712802965842474412t_bool(P,nil_nat),nil_int),nil_int),nil_nat))
=> ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: int,Zs2: list_int,W2: nat,Ws2: list_nat] :
( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_int_nat(size_size_list_int,Ys3) )
=> ( ( aa_list_int_nat(size_size_list_int,Ys3) = aa_list_int_nat(size_size_list_int,Zs2) )
=> ( ( aa_list_int_nat(size_size_list_int,Zs2) = aa_list_nat_nat(size_size_list_nat,Ws2) )
=> ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7542188064555469491t_bool(aa_lis8712802965842474412t_bool(P,Xs2),Ys3),Zs2),Ws2))
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7542188064555469491t_bool(aa_lis8712802965842474412t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_int(Z2,Zs2)),cons_nat(W2,Ws2))) ) ) ) )
=> pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7542188064555469491t_bool(aa_lis8712802965842474412t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_4591_upto__aux__rec,axiom,
! [J: int,I: int,Js: list_int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( upto_aux(I,J,Js) = Js ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).
% upto_aux_rec
tff(fact_4592_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
=> ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
=> ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).
% upto_rec_numeral(4)
tff(fact_4593_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).
% upto_rec_numeral(3)
tff(fact_4594_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).
% upto_rec_numeral(2)
tff(fact_4595_Suc__numeral,axiom,
! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).
% Suc_numeral
tff(fact_4596_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),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_4597_length__upto,axiom,
! [I: int,J: int] : aa_list_int_nat(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_4598_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).
% upto_rec_numeral(1)
tff(fact_4599_num_Osize_I4_J,axiom,
aa_num_nat(size_size_num,one) = zero_zero_nat ).
% num.size(4)
tff(fact_4600_sorted__upto,axiom,
! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).
% sorted_upto
tff(fact_4601_add__One__commute,axiom,
! [N: num] : aa_num_num(plus_plus_num(one),N) = aa_num_num(plus_plus_num(N),one) ).
% add_One_commute
tff(fact_4602_num__induct,axiom,
! [P: fun_num_bool,X: num] :
( pp(aa_num_bool(P,one))
=> ( ! [X2: num] :
( pp(aa_num_bool(P,X2))
=> pp(aa_num_bool(P,aa_num_num(inc,X2))) )
=> pp(aa_num_bool(P,X)) ) ) ).
% num_induct
tff(fact_4603_le__num__One__iff,axiom,
! [X: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),one))
<=> ( X = one ) ) ).
% le_num_One_iff
tff(fact_4604_numerals_I1_J,axiom,
aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).
% numerals(1)
tff(fact_4605_add__One,axiom,
! [X: num] : aa_num_num(plus_plus_num(X),one) = aa_num_num(inc,X) ).
% add_One
tff(fact_4606_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_4607_upto__rec1,axiom,
! [I: int,J: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(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_4608_upto_Osimps,axiom,
! [I: int,J: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(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)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( upto(I,J) = nil_int ) ) ) ).
% upto.simps
tff(fact_4609_upto_Oelims,axiom,
! [X: int,Xa2: int,Y: list_int] :
( ( upto(X,Xa2) = Y )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
=> ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
tff(fact_4610_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),N) ).
% Suc_nat_number_of_add
tff(fact_4611_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).
% neg_zmod_mult_2
tff(fact_4612_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y22) )
<=> ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
tff(fact_4613_semiring__norm_I6_J,axiom,
! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),N)) ).
% semiring_norm(6)
tff(fact_4614_semiring__norm_I2_J,axiom,
aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).
% semiring_norm(2)
tff(fact_4615_num__double,axiom,
! [N: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).
% num_double
tff(fact_4616_add__2__eq__Suc_H,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).
% add_2_eq_Suc'
tff(fact_4617_add__2__eq__Suc,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).
% add_2_eq_Suc
tff(fact_4618_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_4619_add__self__div__2,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).
% add_self_div_2
tff(fact_4620_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
tff(fact_4621_add__self__mod__2,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% add_self_mod_2
tff(fact_4622_half__nonnegative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% half_nonnegative_int_iff
tff(fact_4623_half__negative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% half_negative_int_iff
tff(fact_4624_mod2__gr__0,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).
% mod2_gr_0
tff(fact_4625_verit__eq__simplify_I10_J,axiom,
! [X22: num] : one != aa_num_num(bit0,X22) ).
% verit_eq_simplify(10)
tff(fact_4626_inc_Osimps_I1_J,axiom,
aa_num_num(inc,one) = aa_num_num(bit0,one) ).
% inc.simps(1)
tff(fact_4627_num_Osize_I5_J,axiom,
! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size(5)
tff(fact_4628_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_4629_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_4630_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% power2_nat_le_imp_le
tff(fact_4631_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% power2_nat_le_eq_le
tff(fact_4632_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).
% self_le_ge2_pow
tff(fact_4633_less__2__cases__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
<=> ( ( N = zero_zero_nat )
| ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% less_2_cases_iff
tff(fact_4634_less__2__cases,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
=> ( ( N = zero_zero_nat )
| ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% less_2_cases
tff(fact_4635_nat__induct2,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( pp(aa_nat_bool(P,one_one_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_induct2
tff(fact_4636_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_4637_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).
% diff_le_diff_pow
tff(fact_4638_not__exp__less__eq__0__int,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).
% not_exp_less_eq_0_int
tff(fact_4639_nat__bit__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) )
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_bit_induct
tff(fact_4640_div__2__gt__zero,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% div_2_gt_zero
tff(fact_4641_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% Suc_n_div_2_gt_zero
tff(fact_4642_minus__1__div__exp__eq__int,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% minus_1_div_exp_eq_int
tff(fact_4643_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).
% ex_power_ivl2
tff(fact_4644_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).
% ex_power_ivl1
tff(fact_4645_int__bit__induct,axiom,
! [P: fun_int_bool,K: int] :
( pp(aa_int_bool(P,zero_zero_int))
=> ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
=> ( ! [K3: int] :
( pp(aa_int_bool(P,K3))
=> ( ( K3 != zero_zero_int )
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
=> ( ! [K3: int] :
( pp(aa_int_bool(P,K3))
=> ( ( K3 != aa_int_int(uminus_uminus_int,one_one_int) )
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
=> pp(aa_int_bool(P,K)) ) ) ) ) ).
% int_bit_induct
tff(fact_4646_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).
% pos_zdiv_mult_2
tff(fact_4647_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).
% neg_zdiv_mult_2
tff(fact_4648_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).
% pos_zmod_mult_2
tff(fact_4649_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% unset_bit_nonnegative_int_iff
tff(fact_4650_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% set_bit_nonnegative_int_iff
tff(fact_4651_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% flip_bit_nonnegative_int_iff
tff(fact_4652_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% unset_bit_negative_int_iff
tff(fact_4653_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% set_bit_negative_int_iff
tff(fact_4654_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% flip_bit_negative_int_iff
tff(fact_4655_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] : bit_ri631733984087533419it_int(N,aa_int_int(aa_int_fun_int_int(plus_plus_int,bit_ri631733984087533419it_int(N,K)),bit_ri631733984087533419it_int(N,L))) = bit_ri631733984087533419it_int(N,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).
% signed_take_bit_add
tff(fact_4656_unset__bit__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% unset_bit_nat_def
tff(fact_4657_unset__bit__less__eq,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K)),K)) ).
% unset_bit_less_eq
tff(fact_4658_set__bit__greater__eq,axiom,
! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K))) ).
% set_bit_greater_eq
tff(fact_4659_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),bit_ri631733984087533419it_int(N,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).
% signed_take_bit_int_greater_eq_self_iff
tff(fact_4660_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,bit_ri631733984087533419it_int(N,K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).
% signed_take_bit_int_less_self_iff
tff(fact_4661_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),bit_ri631733984087533419it_int(N,K))) ).
% signed_take_bit_int_greater_eq_minus_exp
tff(fact_4662_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,bit_ri631733984087533419it_int(N,K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)) ) ).
% signed_take_bit_int_less_eq_self_iff
tff(fact_4663_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,bit_ri631733984087533419it_int(N,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).
% signed_take_bit_int_less_eq
tff(fact_4664_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( bit_ri631733984087533419it_int(N,K) = K )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).
% signed_take_bit_int_eq_self_iff
tff(fact_4665_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( bit_ri631733984087533419it_int(N,K) = K ) ) ) ).
% signed_take_bit_int_eq_self
tff(fact_4666_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),bit_ri631733984087533419it_int(N,K))) ) ).
% signed_take_bit_int_greater_eq
tff(fact_4667_binomial__code,axiom,
! [N: nat,K: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).
% binomial_code
tff(fact_4668_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,bit_ri631733984087533419it_int(N,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% signed_take_bit_Suc_minus_bit1
tff(fact_4669_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] : bit_concat_bit(aa_nat_nat(suc,N),K,L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_concat_bit(N,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))) ).
% concat_bit_Suc
tff(fact_4670_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_4671_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
<=> ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
tff(fact_4672_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).
% int_dvd_int_iff
tff(fact_4673_dvd__1__iff__1,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
<=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).
% dvd_1_iff_1
tff(fact_4674_dvd__1__left,axiom,
! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).
% dvd_1_left
tff(fact_4675_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
<=> ( ( K = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% nat_mult_dvd_cancel_disj
tff(fact_4676_zdvd1__eq,axiom,
! [X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
<=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).
% zdvd1_eq
tff(fact_4677_binomial__n__n,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ).
% binomial_n_n
tff(fact_4678_concat__bit__0,axiom,
! [K: int,L: int] : bit_concat_bit(zero_zero_nat,K,L) = L ).
% concat_bit_0
tff(fact_4679_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] : bit_concat_bit(N,K,zero_zero_int) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) ).
% concat_bit_of_zero_2
tff(fact_4680_binomial__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).
% binomial_1
tff(fact_4681_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_4682_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% binomial_eq_0_iff
tff(fact_4683_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).
% binomial_Suc_Suc
tff(fact_4684_semiring__norm_I7_J,axiom,
! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),N)) ).
% semiring_norm(7)
tff(fact_4685_semiring__norm_I9_J,axiom,
! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),N)) ).
% semiring_norm(9)
tff(fact_4686_binomial__n__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).
% binomial_n_0
tff(fact_4687_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).
% dvd_nat_abs_iff
tff(fact_4688_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% nat_abs_dvd_iff
tff(fact_4689_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),bit_concat_bit(N,K,L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).
% concat_bit_nonnegative_iff
tff(fact_4690_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,bit_concat_bit(N,K,L)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).
% concat_bit_negative_iff
tff(fact_4691_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).
% zero_less_binomial_iff
tff(fact_4692_semiring__norm_I3_J,axiom,
! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).
% semiring_norm(3)
tff(fact_4693_semiring__norm_I4_J,axiom,
! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(plus_plus_num(N),one)) ).
% semiring_norm(4)
tff(fact_4694_semiring__norm_I5_J,axiom,
! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).
% semiring_norm(5)
tff(fact_4695_semiring__norm_I8_J,axiom,
! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),one)) ).
% semiring_norm(8)
tff(fact_4696_semiring__norm_I10_J,axiom,
! [M: num,N: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),N)),one)) ).
% semiring_norm(10)
tff(fact_4697_semiring__norm_I16_J,axiom,
! [M: num,N: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),N)),aa_num_num(bit0,aa_num_num(times_times_num(M),N)))) ).
% semiring_norm(16)
tff(fact_4698_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).
% odd_Suc_minus_one
tff(fact_4699_even__diff__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).
% even_diff_nat
tff(fact_4700_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).
% Suc_mod_eq_add3_mod_numeral
tff(fact_4701_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).
% mod_Suc_eq_mod_add3
tff(fact_4702_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).
% Suc_div_eq_add3_div_numeral
tff(fact_4703_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).
% div_Suc_eq_div_add3
tff(fact_4704_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).
% odd_two_times_div_two_nat
tff(fact_4705_zmod__numeral__Bit1,axiom,
! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).
% zmod_numeral_Bit1
tff(fact_4706_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,N),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,bit_ri631733984087533419it_int(N,aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% signed_take_bit_Suc_bit1
tff(fact_4707_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).
% take_bit_nat_less_eq_self
tff(fact_4708_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).
% take_bit_tightened_less_eq_nat
tff(fact_4709_choose__one,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).
% choose_one
tff(fact_4710_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X32) ).
% verit_eq_simplify(14)
tff(fact_4711_verit__eq__simplify_I12_J,axiom,
! [X32: num] : one != aa_num_num(bit1,X32) ).
% verit_eq_simplify(12)
tff(fact_4712_gcd__nat_Oextremum,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).
% gcd_nat.extremum
tff(fact_4713_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
tff(fact_4714_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
<=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
tff(fact_4715_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
tff(fact_4716_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
tff(fact_4717_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).
% dvd_diff_nat
tff(fact_4718_uminus__dvd__conv_I2_J,axiom,
! [D: int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),T))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(uminus_uminus_int,T))) ) ).
% uminus_dvd_conv(2)
tff(fact_4719_uminus__dvd__conv_I1_J,axiom,
! [D: int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),T))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,D)),T)) ) ).
% uminus_dvd_conv(1)
tff(fact_4720_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).
% zdvd_zdiffD
tff(fact_4721_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).
% zdvd_antisym_abs
tff(fact_4722_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ) ) ).
% take_bit_nat_eq
tff(fact_4723_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).
% nat_take_bit_eq
tff(fact_4724_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R: int] : bit_concat_bit(N,K,bit_concat_bit(M,L,R)) = bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N),bit_concat_bit(N,K,L),R) ).
% concat_bit_assoc
tff(fact_4725_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).
% binomial_eq_0
tff(fact_4726_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% binomial_symmetric
tff(fact_4727_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).
% take_bit_tightened_less_eq_int
tff(fact_4728_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% take_bit_int_less_eq_self_iff
tff(fact_4729_take__bit__nonnegative,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ).
% take_bit_nonnegative
tff(fact_4730_not__take__bit__negative,axiom,
! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),zero_zero_int)) ).
% not_take_bit_negative
tff(fact_4731_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% take_bit_int_greater_self_iff
tff(fact_4732_choose__mult__lemma,axiom,
! [M: nat,R: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),M)) ).
% choose_mult_lemma
tff(fact_4733_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] : Y != aa_num_num(bit0,X23)
=> ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).
% num.exhaust
tff(fact_4734_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).
% binomial_le_pow
tff(fact_4735_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).
% dvd_pos_nat
tff(fact_4736_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).
% nat_dvd_not_less
tff(fact_4737_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% dvd_minus_self
tff(fact_4738_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
tff(fact_4739_zdvd__not__zless,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).
% zdvd_not_zless
tff(fact_4740_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% less_eq_dvd_minus
tff(fact_4741_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).
% dvd_diffD1
tff(fact_4742_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).
% dvd_diffD
tff(fact_4743_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N)))
=> ( ( K != zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).
% zdvd_mult_cancel
tff(fact_4744_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),T))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),T))) ) ) ).
% zdvd_mono
tff(fact_4745_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D5: nat,X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),B))
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D5) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D5) ) ) ) ).
% bezout_add_nat
tff(fact_4746_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
=> ? [X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y2)),D) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) ) ) ) ) ) ).
% bezout_lemma_nat
tff(fact_4747_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D5: nat,X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),B))
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = D5 )
| ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = D5 ) ) ) ).
% bezout1_nat
tff(fact_4748_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K),M))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).
% zdvd_reduce
tff(fact_4749_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T))) ) ) ).
% zdvd_period
tff(fact_4750_inc_Osimps_I3_J,axiom,
! [X: num] : aa_num_num(inc,aa_num_num(bit1,X)) = aa_num_num(bit0,aa_num_num(inc,X)) ).
% inc.simps(3)
tff(fact_4751_inc_Osimps_I2_J,axiom,
! [X: num] : aa_num_num(inc,aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).
% inc.simps(2)
tff(fact_4752_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
tff(fact_4753_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).
% zero_less_binomial
tff(fact_4754_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_4755_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).
% choose_mult
tff(fact_4756_binomial__absorb__comp,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).
% binomial_absorb_comp
tff(fact_4757_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != zero_zero_int )
=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),one_one_int) ) ) ).
% take_bit_decr_eq
tff(fact_4758_eval__nat__numeral_I3_J,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).
% eval_nat_numeral(3)
tff(fact_4759_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).
% dvd_imp_le
tff(fact_4760_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% dvd_mult_cancel
tff(fact_4761_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% nat_mult_dvd_cancel1
tff(fact_4762_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D5: nat,X2: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),B))
& ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D5) ) ) ) ).
% bezout_add_strong_nat
tff(fact_4763_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).
% zdvd_imp_le
tff(fact_4764_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).
% mod_greater_zero_iff_not_dvd
tff(fact_4765_dvd__imp__le__int,axiom,
! [I: int,D: int] :
( ( I != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).
% dvd_imp_le_int
tff(fact_4766_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).
% mod_eq_dvd_iff_nat
tff(fact_4767_dvd__fact,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).
% dvd_fact
tff(fact_4768_even__nat__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).
% even_nat_iff
tff(fact_4769_gcd__nat_Oordering__top__axioms,axiom,
pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_cd),zero_zero_nat)) ).
% gcd_nat.ordering_top_axioms
tff(fact_4770_binomial__absorption,axiom,
! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).
% binomial_absorption
tff(fact_4771_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% odd_mod_4_div_2
tff(fact_4772_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_4773_Suc3__eq__add__3,axiom,
! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).
% Suc3_eq_add_3
tff(fact_4774_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).
% binomial_fact_lemma
tff(fact_4775_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
<=> ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
tff(fact_4776_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
<=> ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
tff(fact_4777_even__even__mod__4__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).
% even_even_mod_4_iff
tff(fact_4778_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q2)))) ) ) ) ).
% dvd_minus_add
tff(fact_4779_mod__nat__eqI,axiom,
! [R: nat,N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R ) ) ) ) ).
% mod_nat_eqI
tff(fact_4780_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M))
<=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).
% zdvd_mult_cancel1
tff(fact_4781_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
| ( ( L = zero_zero_int )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).
% mod_int_pos_iff
tff(fact_4782_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_dvd_imp_le
tff(fact_4783_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),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_4784_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_4785_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).
% choose_reduce_nat
tff(fact_4786_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% times_binomial_minus1_eq
tff(fact_4787_binomial__le__pow2,axiom,
! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% binomial_le_pow2
tff(fact_4788_odd__pos,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% odd_pos
tff(fact_4789_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).
% binomial_altdef_nat
tff(fact_4790_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).
% Suc_mod_eq_add3_mod
tff(fact_4791_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).
% Suc_div_eq_add3_div
tff(fact_4792_even__diff__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).
% even_diff_iff
tff(fact_4793_even__add__abs__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).
% even_add_abs_iff
tff(fact_4794_even__abs__add__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).
% even_abs_add_iff
tff(fact_4795_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% dvd_power_iff_le
tff(fact_4796_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)) ) ).
% take_bit_nat_less_self_iff
tff(fact_4797_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).
% binomial_addition_formula
tff(fact_4798_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).
% take_bit_int_less_self_iff
tff(fact_4799_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).
% take_bit_int_greater_eq_self_iff
tff(fact_4800_choose__two,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% choose_two
tff(fact_4801_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).
% take_bit_int_eq_self_iff
tff(fact_4802_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K ) ) ) ).
% take_bit_int_eq_self
tff(fact_4803_signed__take__bit__eq__take__bit__shift,axiom,
! [N: nat,K: int] : bit_ri631733984087533419it_int(N,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% signed_take_bit_eq_take_bit_shift
tff(fact_4804_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ) ) ).
% take_bit_incr_eq
tff(fact_4805_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).
% take_bit_int_less_eq
tff(fact_4806_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).
% take_bit_int_greater_eq
tff(fact_4807_even__mod__4__div__2,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% even_mod_4_div_2
tff(fact_4808_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).
% take_bit_minus_small_eq
tff(fact_4809_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : 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,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_4810_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] : 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,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_4811_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa_int_int(aa_nat_fun_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(aa_nat_fun_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_4812_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_4813_num_Osize__gen_I2_J,axiom,
! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size_gen(2)
tff(fact_4814_nat__of__bool,axiom,
! [P: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P) ).
% nat_of_bool
tff(fact_4815_Suc__0__mod__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).
% Suc_0_mod_eq
tff(fact_4816_pred__numeral__simps_I1_J,axiom,
aa_num_nat(pred_numeral,one) = zero_zero_nat ).
% pred_numeral_simps(1)
tff(fact_4817_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
<=> ( aa_num_nat(pred_numeral,K) = N ) ) ).
% eq_numeral_Suc
tff(fact_4818_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
<=> ( N = aa_num_nat(pred_numeral,K) ) ) ).
% Suc_eq_numeral
tff(fact_4819_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_4820_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).
% less_numeral_Suc
tff(fact_4821_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).
% less_Suc_numeral
tff(fact_4822_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_4823_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).
% le_numeral_Suc
tff(fact_4824_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).
% le_Suc_numeral
tff(fact_4825_diff__numeral__Suc,axiom,
! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).
% diff_numeral_Suc
tff(fact_4826_diff__Suc__numeral,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).
% diff_Suc_numeral
tff(fact_4827_take__bit__of__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_Suc_0
tff(fact_4828_dvd__antisym,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
=> ( M = N ) ) ) ).
% dvd_antisym
tff(fact_4829_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
tff(fact_4830_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
tff(fact_4831_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% gcd_nat.strict_implies_order
tff(fact_4832_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
tff(fact_4833_gcd__nat_Oorder__iff__strict,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
| ( A = B ) ) ) ).
% gcd_nat.order_iff_strict
tff(fact_4834_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).
% gcd_nat.strict_iff_not
tff(fact_4835_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
tff(fact_4836_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
& ( B != C ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
tff(fact_4837_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
& ( B != C ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
tff(fact_4838_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( A = B ) ) ) ).
% gcd_nat.antisym
tff(fact_4839_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A))
& ( A != A ) ) ).
% gcd_nat.irrefl
tff(fact_4840_gcd__nat_Oeq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).
% gcd_nat.eq_iff
tff(fact_4841_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% gcd_nat.trans
tff(fact_4842_gcd__nat_Orefl,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).
% gcd_nat.refl
tff(fact_4843_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
& ( B != A ) ) ) ).
% gcd_nat.asym
tff(fact_4844_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_4845_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_4846_num_Osize__gen_I1_J,axiom,
aa_num_nat(size_num,one) = zero_zero_nat ).
% num.size_gen(1)
tff(fact_4847_set__decode__0,axiom,
! [X: nat] :
( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).
% set_decode_0
tff(fact_4848_set__decode__def,axiom,
! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ce,X)) ).
% set_decode_def
tff(fact_4849_and__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = zero_zero_int )
| ( L = zero_zero_int ) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = zero_zero_int ) )
& ( ~ ( ( K = zero_zero_int )
| ( L = zero_zero_int ) )
=> ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = L ) )
& ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = K ) )
& ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).
% and_int_unfold
tff(fact_4850_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))) ) ).
% take_bit_eq_mask_iff_exp_dvd
tff(fact_4851_modulo__int__def,axiom,
! [L: int,K: int] :
( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) )
& ( ( L != zero_zero_int )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).
% modulo_int_def
tff(fact_4852_mask__nat__positive__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% mask_nat_positive_iff
tff(fact_4853_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).
% and_nonnegative_int_iff
tff(fact_4854_and__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).
% and_negative_int_iff
tff(fact_4855_sgn__mult__dvd__iff,axiom,
! [R: int,L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),L)),K))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
& ( ( R = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
tff(fact_4856_mult__sgn__dvd__iff,axiom,
! [L: int,R: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R))),K))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
& ( ( R = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
tff(fact_4857_dvd__sgn__mult__iff,axiom,
! [L: int,R: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),K)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
| ( R = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
tff(fact_4858_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R))))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
| ( R = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
tff(fact_4859_and__minus__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).
% and_minus_numerals(2)
tff(fact_4860_and__minus__numerals_I6_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).
% and_minus_numerals(6)
tff(fact_4861_and__minus__numerals_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).
% and_minus_numerals(5)
tff(fact_4862_and__minus__numerals_I1_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).
% and_minus_numerals(1)
tff(fact_4863_less__eq__mask,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).
% less_eq_mask
tff(fact_4864_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% subset_decode_imp_le
tff(fact_4865_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z)) ) ) ).
% AND_upper2'
tff(fact_4866_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).
% AND_upper1'
tff(fact_4867_AND__upper2,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Y)) ) ).
% AND_upper2
tff(fact_4868_AND__upper1,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),X)) ) ).
% AND_upper1
tff(fact_4869_AND__lower,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y))) ) ).
% AND_lower
tff(fact_4870_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L2: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).
% int_sgnE
tff(fact_4871_mask__nonnegative__int,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).
% mask_nonnegative_int
tff(fact_4872_not__mask__negative__int,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).
% not_mask_negative_int
tff(fact_4873_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z)) ) ) ).
% AND_upper2''
tff(fact_4874_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).
% AND_upper1''
tff(fact_4875_and__less__eq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K)) ) ).
% and_less_eq
tff(fact_4876_sgn__mod,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
=> ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).
% sgn_mod
tff(fact_4877_less__mask,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).
% less_mask
tff(fact_4878_zsgn__def,axiom,
! [I: int] :
( ( ( I = zero_zero_int )
=> ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
& ( ( I != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
=> ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
=> ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).
% zsgn_def
tff(fact_4879_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).
% div_sgn_abs_cancel
tff(fact_4880_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
<=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).
% take_bit_eq_mask_iff
tff(fact_4881_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_bool_int(zero_n2684676970156552555ol_int,fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).
% div_noneq_sgn_abs
tff(fact_4882_and__int__rec,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% and_int_rec
tff(fact_4883_mask__half__int,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).
% mask_half_int
tff(fact_4884_mask__nat__def,axiom,
! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).
% mask_nat_def
tff(fact_4885_mask__int__def,axiom,
! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).
% mask_int_def
tff(fact_4886_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
& ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).
% divide_int_unfold
tff(fact_4887_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
& ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).
% modulo_int_unfold
tff(fact_4888_divide__int__def,axiom,
! [L: int,K: int] :
( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) )
& ( ( L != zero_zero_int )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).
% divide_int_def
tff(fact_4889_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)) ).
% take_bit_Suc_from_most
tff(fact_4890_Suc__0__xor__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% Suc_0_xor_eq
tff(fact_4891_xor__Suc__0__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% xor_Suc_0_eq
tff(fact_4892_horner__sum__of__bool__2__less,axiom,
! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_list_bool_nat(size_size_list_bool,Bs)))) ).
% horner_sum_of_bool_2_less
tff(fact_4893_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).
% xor_nonnegative_int_iff
tff(fact_4894_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% push_bit_nonnegative_int_iff
tff(fact_4895_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int))
<=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).
% xor_negative_int_iff
tff(fact_4896_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% push_bit_negative_int_iff
tff(fact_4897_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] : bit_concat_bit(N,zero_zero_int,L) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),L) ).
% concat_bit_of_zero_1
tff(fact_4898_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),bit_ri631733984087533419it_int(N,K)))
<=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).
% signed_take_bit_nonnegative_iff
tff(fact_4899_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,bit_ri631733984087533419it_int(N,K)),zero_zero_int))
<=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).
% signed_take_bit_negative_iff
tff(fact_4900_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_4901_and__nat__numerals_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
% and_nat_numerals(3)
tff(fact_4902_push__bit__of__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).
% push_bit_of_Suc_0
tff(fact_4903_xor__nat__numerals_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).
% xor_nat_numerals(4)
tff(fact_4904_xor__nat__numerals_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).
% xor_nat_numerals(3)
tff(fact_4905_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_4906_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_4907_and__nat__numerals_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% and_nat_numerals(4)
tff(fact_4908_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_4909_and__Suc__0__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% and_Suc_0_eq
tff(fact_4910_Suc__0__and__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Suc_0_and_eq
tff(fact_4911_flip__bit__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).
% flip_bit_nat_def
tff(fact_4912_xor__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% xor_nat_def
tff(fact_4913_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% bit_push_bit_iff_int
tff(fact_4914_flip__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).
% flip_bit_int_def
tff(fact_4915_XOR__lower,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y))) ) ) ).
% XOR_lower
tff(fact_4916_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
<=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).
% bit_not_int_iff'
tff(fact_4917_and__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% and_nat_def
tff(fact_4918_concat__bit__eq,axiom,
! [N: nat,K: int,L: int] : bit_concat_bit(N,K,L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),L)) ).
% concat_bit_eq
tff(fact_4919_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K))) ) ) ).
% bit_imp_take_bit_positive
tff(fact_4920_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,bit_concat_bit(M,K,L)),N))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
& pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).
% bit_concat_bit_iff
tff(fact_4921_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M6))
=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M6))
<=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
<=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).
% int_bit_bound
tff(fact_4922_push__bit__minus__one,axiom,
! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% push_bit_minus_one
tff(fact_4923_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).
% XOR_upper
tff(fact_4924_and__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
& ( ~ ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).
% and_nat_unfold
tff(fact_4925_and__nat__rec,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,fconj(fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% and_nat_rec
tff(fact_4926_xor__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
& ( ( M != zero_zero_nat )
=> ( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).
% xor_nat_unfold
tff(fact_4927_xor__int__rec,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fNot(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% xor_int_rec
tff(fact_4928_xor__nat__rec,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,fNot(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% xor_nat_rec
tff(fact_4929_set__bit__eq,axiom,
! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,fNot(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).
% set_bit_eq
tff(fact_4930_xor__int__unfold,axiom,
! [K: int,L: int] :
( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
& ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
& ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( ( K = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = L ) )
& ( ( K != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = K ) )
& ( ( L != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
tff(fact_4931_mask__eq__sum__exp__nat,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) ).
% mask_eq_sum_exp_nat
tff(fact_4932_not__nonnegative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% not_nonnegative_int_iff
tff(fact_4933_not__negative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% not_negative_int_iff
tff(fact_4934_bit__Suc__0__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
<=> ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
tff(fact_4935_not__bit__Suc__0__Suc,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).
% not_bit_Suc_0_Suc
tff(fact_4936_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q2)),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% bit_push_bit_iff_nat
tff(fact_4937_not__bit__Suc__0__numeral,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).
% not_bit_Suc_0_numeral
tff(fact_4938_not__int__def,axiom,
! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).
% not_int_def
tff(fact_4939_and__not__numerals_I1_J,axiom,
aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).
% and_not_numerals(1)
tff(fact_4940_unset__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int))) ).
% unset_bit_int_def
tff(fact_4941_and__not__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).
% and_not_numerals(2)
tff(fact_4942_and__not__numerals_I4_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).
% and_not_numerals(4)
tff(fact_4943_bit__nat__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ) ).
% bit_nat_iff
tff(fact_4944_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),N))
<=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N)) ) ).
% bit_minus_int_iff
tff(fact_4945_and__not__numerals_I7_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).
% and_not_numerals(7)
tff(fact_4946_and__not__numerals_I3_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).
% and_not_numerals(3)
tff(fact_4947_and__not__numerals_I8_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% and_not_numerals(8)
tff(fact_4948_not__int__rec,axiom,
! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% not_int_rec
tff(fact_4949_atMost__atLeast0,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N) ).
% atMost_atLeast0
tff(fact_4950_all__nat__less,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
=> pp(aa_nat_bool(P,M5)) )
<=> ! [X6: nat] :
( pp(aa_set_nat_bool(member_nat(X6),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
=> pp(aa_nat_bool(P,X6)) ) ) ).
% all_nat_less
tff(fact_4951_ex__nat__less,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
& pp(aa_nat_bool(P,M5)) )
<=> ? [X6: nat] :
( pp(aa_set_nat_bool(member_nat(X6),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
& pp(aa_nat_bool(P,X6)) ) ) ).
% ex_nat_less
tff(fact_4952_sum__choose__upper,axiom,
! [M: nat,N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cf,M),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).
% sum_choose_upper
tff(fact_4953_sum__choose__lower,axiom,
! [R: nat,N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cg,R),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),N))),N) ).
% sum_choose_lower
tff(fact_4954_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ch,N),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).
% choose_rising_sum(1)
tff(fact_4955_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ch,N),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).
% choose_rising_sum(2)
tff(fact_4956_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ci(M),N),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).
% sum_choose_diagonal
tff(fact_4957_vandermonde,axiom,
! [M: nat,N: nat,R: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_cj(M),N),R),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R) ).
% vandermonde
tff(fact_4958_choose__row__sum,axiom,
! [N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).
% choose_row_sum
tff(fact_4959_binomial,axiom,
! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ck(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).
% binomial
tff(fact_4960_choose__square__sum,axiom,
! [N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cl,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).
% choose_square_sum
tff(fact_4961_gauss__sum__nat,axiom,
! [N: nat] : groups3542108847815614940at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum_nat
tff(fact_4962_binomial__r__part__sum,axiom,
! [M: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).
% binomial_r_part_sum
tff(fact_4963_choose__linear__sum,axiom,
! [N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cm,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).
% choose_linear_sum
tff(fact_4964_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cn(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% arith_series_nat
tff(fact_4965_Sum__Icc__nat,axiom,
! [M: nat,N: nat] : groups3542108847815614940at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Sum_Icc_nat
tff(fact_4966_Sum__Icc__int,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
=> ( groups4538972089207619220nt_int(aTP_Lamm_bu,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% Sum_Icc_int
tff(fact_4967_Sum__Ico__nat,axiom,
! [M: nat,N: nat] : groups3542108847815614940at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Sum_Ico_nat
tff(fact_4968_sum__power2,axiom,
! [K: 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_4969_prod__Suc__Suc__fact,axiom,
! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).
% prod_Suc_Suc_fact
tff(fact_4970_prod__Suc__fact,axiom,
! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).
% prod_Suc_fact
tff(fact_4971_all__nat__less__eq,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
=> pp(aa_nat_bool(P,M5)) )
<=> ! [X6: nat] :
( pp(aa_set_nat_bool(member_nat(X6),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
=> pp(aa_nat_bool(P,X6)) ) ) ).
% all_nat_less_eq
tff(fact_4972_ex__nat__less__eq,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
& pp(aa_nat_bool(P,M5)) )
<=> ? [X6: nat] :
( pp(aa_set_nat_bool(member_nat(X6),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
& pp(aa_nat_bool(P,X6)) ) ) ).
% ex_nat_less_eq
tff(fact_4973_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_4974_bset_I1_J,axiom,
! [D3: int,B3: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,B3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,B3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X2))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))) ) )
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ( pp(aa_int_bool(P,X3))
& pp(aa_int_bool(Q,X3)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)))
& pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) ) ) ) ) ).
% bset(1)
tff(fact_4975_bset_I2_J,axiom,
! [D3: int,B3: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,B3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,B3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X2))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))) ) )
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ( pp(aa_int_bool(P,X3))
| pp(aa_int_bool(Q,X3)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)))
| pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) ) ) ) ) ).
% bset(2)
tff(fact_4976_aset_I1_J,axiom,
! [D3: int,A3: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,A3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,A3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X2))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))) ) )
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ( pp(aa_int_bool(P,X3))
& pp(aa_int_bool(Q,X3)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)))
& pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) ) ) ) ) ).
% aset(1)
tff(fact_4977_aset_I2_J,axiom,
! [D3: int,A3: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,A3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,A3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X2))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))) ) )
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ( pp(aa_int_bool(P,X3))
| pp(aa_int_bool(Q,X3)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)))
| pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) ) ) ) ) ).
% aset(2)
tff(fact_4978_bset_I9_J,axiom,
! [D: int,D3: int,B3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T)))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T))) ) ) ) ).
% bset(9)
tff(fact_4979_bset_I10_J,axiom,
! [D: int,D3: int,B3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T)))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T))) ) ) ) ).
% bset(10)
tff(fact_4980_aset_I9_J,axiom,
! [D: int,D3: int,A3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T)))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T))) ) ) ) ).
% aset(9)
tff(fact_4981_aset_I10_J,axiom,
! [D: int,D3: int,A3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T)))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T))) ) ) ) ).
% aset(10)
tff(fact_4982_periodic__finite__ex,axiom,
! [D: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
=> ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
<=> ? [X6: int] :
( member_int(X6,set_or1266510415728281911st_int(one_one_int,D))
& pp(aa_int_bool(P,X6)) ) ) ) ) ).
% periodic_finite_ex
tff(fact_4983_bset_I3_J,axiom,
! [D3: int,T: int,B3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ( X3 = T )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3) = T ) ) ) ) ) ).
% bset(3)
tff(fact_4984_bset_I4_J,axiom,
! [D3: int,T: int,B3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(T,B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( ( X3 != T )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3) != T ) ) ) ) ) ).
% bset(4)
tff(fact_4985_bset_I5_J,axiom,
! [D3: int,B3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T)) ) ) ) ).
% bset(5)
tff(fact_4986_bset_I7_J,axiom,
! [D3: int,T: int,B3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(T,B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) ) ) ) ).
% bset(7)
tff(fact_4987_aset_I3_J,axiom,
! [D3: int,T: int,A3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ( X3 = T )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3) = T ) ) ) ) ) ).
% aset(3)
tff(fact_4988_aset_I4_J,axiom,
! [D3: int,T: int,A3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(T,A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( ( X3 != T )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3) != T ) ) ) ) ) ).
% aset(4)
tff(fact_4989_aset_I5_J,axiom,
! [D3: int,T: int,A3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(T,A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T)) ) ) ) ) ).
% aset(5)
tff(fact_4990_aset_I7_J,axiom,
! [D3: int,A3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) ) ) ).
% aset(7)
tff(fact_4991_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),groups708209901874060359at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).
% fact_eq_fact_times
tff(fact_4992_bset_I6_J,axiom,
! [D3: int,B3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3)),T)) ) ) ) ).
% bset(6)
tff(fact_4993_bset_I8_J,axiom,
! [D3: int,T: int,B3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,B3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) ) ) ) ).
% bset(8)
tff(fact_4994_aset_I6_J,axiom,
! [D3: int,T: int,A3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A3)
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3)),T)) ) ) ) ) ).
% aset(6)
tff(fact_4995_aset_I8_J,axiom,
! [D3: int,A3: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X3: int] :
( ! [Xa3: int] :
( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb3: int] :
( member_int(Xb3,A3)
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) ) ) ).
% aset(8)
tff(fact_4996_cppi,axiom,
! [D3: int,P: fun_int_bool,P2: fun_int_bool,A3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X2))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,A3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))) ) )
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(P2,X2))
<=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
<=> ( ? [X6: int] :
( member_int(X6,set_or1266510415728281911st_int(one_one_int,D3))
& pp(aa_int_bool(P2,X6)) )
| ? [X6: int] :
( member_int(X6,set_or1266510415728281911st_int(one_one_int,D3))
& ? [Xa4: int] :
( member_int(Xa4,A3)
& pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X6))) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_4997_cpmi,axiom,
! [D3: int,P: fun_int_bool,P2: fun_int_bool,B3: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( ? [Z3: int] :
! [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z3))
=> ( pp(aa_int_bool(P,X2))
<=> pp(aa_int_bool(P2,X2)) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
=> ! [Xb2: int] :
( member_int(Xb2,B3)
=> ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))) ) )
=> ( ! [X2: int,K3: int] :
( pp(aa_int_bool(P2,X2))
<=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
<=> ( ? [X6: int] :
( member_int(X6,set_or1266510415728281911st_int(one_one_int,D3))
& pp(aa_int_bool(P2,X6)) )
| ? [X6: int] :
( member_int(X6,set_or1266510415728281911st_int(one_one_int,D3))
& ? [Xa4: int] :
( member_int(Xa4,B3)
& pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X6))) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_4998_fact__div__fact,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).
% fact_div_fact
tff(fact_4999_integer__of__int__code,axiom,
! [K: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( ( ( K = zero_zero_int )
=> ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
& ( ( K != zero_zero_int )
=> ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_Code_integer)) ) ) ) ) ) ).
% integer_of_int_code
tff(fact_5000_or__int__rec,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fdisj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% or_int_rec
tff(fact_5001_integer__of__int__eq__of__int,axiom,
code_integer_of_int = ring_18347121197199848620nteger ).
% integer_of_int_eq_of_int
tff(fact_5002_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))) ).
% finite_atLeastAtMost
tff(fact_5003_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))) ).
% finite_atLeastLessThan
tff(fact_5004_finite__atMost,axiom,
! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K))) ).
% finite_atMost
tff(fact_5005_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).
% or_nonnegative_int_iff
tff(fact_5006_or__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).
% or_negative_int_iff
tff(fact_5007_or__nat__numerals_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).
% or_nat_numerals(4)
tff(fact_5008_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_5009_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_5010_or__nat__numerals_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).
% or_nat_numerals(3)
tff(fact_5011_or__minus__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).
% or_minus_numerals(2)
tff(fact_5012_or__minus__numerals_I6_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).
% or_minus_numerals(6)
tff(fact_5013_and__minus__minus__numerals,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).
% and_minus_minus_numerals
tff(fact_5014_or__minus__minus__numerals,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).
% or_minus_minus_numerals
tff(fact_5015_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5016_or__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% or_nat_def
tff(fact_5017_less__eq__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X)) ) ).
% less_eq_integer.abs_eq
tff(fact_5018_less__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X)) ) ).
% less_integer.abs_eq
tff(fact_5019_one__integer__def,axiom,
one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).
% one_integer_def
tff(fact_5020_times__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa2),X)) ).
% times_integer.abs_eq
tff(fact_5021_uminus__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,X)) ).
% uminus_integer.abs_eq
tff(fact_5022_zero__integer__def,axiom,
zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).
% zero_integer_def
tff(fact_5023_sgn__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).
% sgn_integer_code
tff(fact_5024_abs__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = K ) ) ) ).
% abs_integer_code
tff(fact_5025_less__integer__code_I1_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).
% less_integer_code(1)
tff(fact_5026_minus__integer__code_I2_J,axiom,
! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).
% minus_integer_code(2)
tff(fact_5027_minus__integer__code_I1_J,axiom,
! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,K),zero_z3403309356797280102nteger) = K ).
% minus_integer_code(1)
tff(fact_5028_uminus__integer__code_I1_J,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% uminus_integer_code(1)
tff(fact_5029_or__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa2),X)) ).
% or_integer.abs_eq
tff(fact_5030_plus__integer__code_I2_J,axiom,
! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),L) = L ).
% plus_integer_code(2)
tff(fact_5031_plus__integer__code_I1_J,axiom,
! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),zero_z3403309356797280102nteger) = K ).
% plus_integer_code(1)
tff(fact_5032_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_5033_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_5034_plus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X)) ).
% plus_integer.abs_eq
tff(fact_5035_minus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X)) ).
% minus_integer.abs_eq
tff(fact_5036_abs__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(abs_abs_int,X)) ).
% abs_integer.abs_eq
tff(fact_5037_divide__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),X)) ).
% divide_integer.abs_eq
tff(fact_5038_modulo__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa2),X)) ).
% modulo_integer.abs_eq
tff(fact_5039_take__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Xa2),X)) ).
% take_bit_integer.abs_eq
tff(fact_5040_not__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(bit_ri7919022796975470100ot_int,X)) ).
% not_integer.abs_eq
tff(fact_5041_and__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa2),X)) ).
% and_integer.abs_eq
tff(fact_5042_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),N)) )
=> pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).
% bounded_nat_set_is_finite
tff(fact_5043_finite__nat__set__iff__bounded,axiom,
! [N4: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,N4))
<=> ? [M5: nat] :
! [X6: nat] :
( pp(aa_set_nat_bool(member_nat(X6),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),M5)) ) ) ).
% finite_nat_set_iff_bounded
tff(fact_5044_sgn__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(sgn_sgn_int,X)) ).
% sgn_integer.abs_eq
tff(fact_5045_finite__nat__set__iff__bounded__le,axiom,
! [N4: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,N4))
<=> ? [M5: nat] :
! [X6: nat] :
( pp(aa_set_nat_bool(member_nat(X6),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),M5)) ) ) ).
% finite_nat_set_iff_bounded_le
tff(fact_5046_bit__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod2498268374540730821t_bool(bit_se9216721137139052372nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_fun_nat_bool(bit_se1146084159140164899it_int,X) ).
% bit_integer.abs_eq
tff(fact_5047_xor__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa2),X)) ).
% xor_integer.abs_eq
tff(fact_5048_push__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se7788150548672797655nteger,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Xa2),X)) ).
% push_bit_integer.abs_eq
tff(fact_5049_mask__integer_Oabs__eq,axiom,
! [X: nat] : aa_nat_Code_integer(bit_se2119862282449309892nteger,X) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(bit_se2000444600071755411sk_int,X)) ).
% mask_integer.abs_eq
tff(fact_5050_unset__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Xa2),X)) ).
% unset_bit_integer.abs_eq
tff(fact_5051_set__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Xa2),X)) ).
% set_bit_integer.abs_eq
tff(fact_5052_flip__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Xa2),X)) ).
% flip_bit_integer.abs_eq
tff(fact_5053_finite__M__bounded__by__nat,axiom,
! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_co(P),I)))) ).
% finite_M_bounded_by_nat
tff(fact_5054_finite__less__ub,axiom,
! [F: fun_nat_nat,U: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cp(F),U)))) ) ).
% finite_less_ub
tff(fact_5055_or__greater__eq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))) ) ).
% or_greater_eq
tff(fact_5056_OR__lower,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y))) ) ) ).
% OR_lower
tff(fact_5057_plus__and__or,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) ).
% plus_and_or
tff(fact_5058_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
<=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_cq),Nat)) ) ).
% nat.disc_eq_case(1)
tff(fact_5059_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
<=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_cr),Nat)) ) ).
% nat.disc_eq_case(2)
tff(fact_5060_set__bit__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).
% set_bit_nat_def
tff(fact_5061_or__not__numerals_I1_J,axiom,
aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).
% or_not_numerals(1)
tff(fact_5062_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_5063_prod__int__eq,axiom,
! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_bu,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).
% prod_int_eq
tff(fact_5064_set__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),one_one_int)) ).
% set_bit_int_def
tff(fact_5065_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
<=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).
% less_eq_nat.simps(2)
tff(fact_5066_finite__divisors__nat,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cs,M)))) ) ).
% finite_divisors_nat
tff(fact_5067_subset__eq__atLeast0__atMost__finite,axiom,
! [N4: set_nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
=> pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).
% subset_eq_atLeast0_atMost_finite
tff(fact_5068_subset__eq__atLeast0__lessThan__finite,axiom,
! [N4: set_nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
=> pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).
% subset_eq_atLeast0_lessThan_finite
tff(fact_5069_or__not__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).
% or_not_numerals(2)
tff(fact_5070_or__not__numerals_I4_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).
% or_not_numerals(4)
tff(fact_5071_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_bu,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_5072_diff__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bt),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).
% diff_Suc
tff(fact_5073_or__not__numerals_I3_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).
% or_not_numerals(3)
tff(fact_5074_or__not__numerals_I7_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).
% or_not_numerals(7)
tff(fact_5075_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).
% OR_upper
tff(fact_5076_or__not__numerals_I5_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% or_not_numerals(5)
tff(fact_5077_or__Suc__0__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% or_Suc_0_eq
tff(fact_5078_Suc__0__or__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% Suc_0_or_eq
tff(fact_5079_or__not__numerals_I9_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% or_not_numerals(9)
tff(fact_5080_or__not__numerals_I8_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% or_not_numerals(8)
tff(fact_5081_or__nat__rec,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,fdisj(fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),fNot(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% or_nat_rec
tff(fact_5082_finite__Collect__le__nat,axiom,
! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,K)))) ).
% finite_Collect_le_nat
tff(fact_5083_finite__Collect__less__nat,axiom,
! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,K)))) ).
% finite_Collect_less_nat
tff(fact_5084_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : finite_finite_int(set_or1266510415728281911st_int(L,U)) ).
% finite_atLeastAtMost_int
tff(fact_5085_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : finite_finite_int(set_or4662586982721622107an_int(L,U)) ).
% finite_atLeastLessThan_int
tff(fact_5086_finite__interval__int1,axiom,
! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ct(A),B))) ).
% finite_interval_int1
tff(fact_5087_finite__interval__int4,axiom,
! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_cu(A),B))) ).
% finite_interval_int4
tff(fact_5088_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or8404916559141939852nteger(L,U)) ).
% finite_atLeastLessThan_integer
tff(fact_5089_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or189985376899183464nteger(L,U)) ).
% finite_atLeastAtMost_integer
tff(fact_5090_finite__interval__int3,axiom,
! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_cv(A),B))) ).
% finite_interval_int3
tff(fact_5091_finite__interval__int2,axiom,
! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_cw(A),B))) ).
% finite_interval_int2
tff(fact_5092_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : finite_finite_int(set_or4662586982721622107an_int(zero_zero_int,U)) ).
% finite_atLeastZeroLessThan_int
tff(fact_5093_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : finite6017078050557962740nteger(set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).
% finite_atLeastZeroLessThan_integer
tff(fact_5094_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero_int )
=> finite_finite_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_cx,I))) ) ).
% finite_divisors_int
tff(fact_5095_or__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
| ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
& ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
| ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
=> ( ( ( K = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = L ) )
& ( ( K != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = K ) )
& ( ( L != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).
% or_int_unfold
tff(fact_5096_even__set__encode__iff,axiom,
! [A3: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),nat_set_encode(A3)))
<=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) ) ) ).
% even_set_encode_iff
tff(fact_5097_Code__Numeral_Opositive__def,axiom,
code_positive = numera6620942414471956472nteger ).
% Code_Numeral.positive_def
tff(fact_5098_or__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
& ( ( M != zero_zero_nat )
=> ( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).
% or_nat_unfold
tff(fact_5099_max__0R,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).
% max_0R
tff(fact_5100_max__0L,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).
% max_0L
tff(fact_5101_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_5102_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_5103_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_5104_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_5105_max__Suc__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).
% max_Suc_Suc
tff(fact_5106_max__numeral__Suc,axiom,
! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).
% max_numeral_Suc
tff(fact_5107_max__Suc__numeral,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).
% max_Suc_numeral
tff(fact_5108_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).
% nat_mult_max_right
tff(fact_5109_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).
% nat_mult_max_left
tff(fact_5110_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).
% nat_add_max_left
tff(fact_5111_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).
% nat_add_max_right
tff(fact_5112_nat__minus__add__max,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).
% nat_minus_add_max
tff(fact_5113_max__Suc1,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_cy,N)),M) ).
% max_Suc1
tff(fact_5114_max__Suc2,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_cz,N)),M) ).
% max_Suc2
tff(fact_5115_set__encode__inf,axiom,
! [A3: set_nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
=> ( nat_set_encode(A3) = zero_zero_nat ) ) ).
% set_encode_inf
tff(fact_5116_infinite__int__iff__unbounded__le,axiom,
! [S3: set_int] :
( ~ finite_finite_int(S3)
<=> ! [M5: int] :
? [N3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N3)))
& member_int(N3,S3) ) ) ).
% infinite_int_iff_unbounded_le
tff(fact_5117_set__encode__insert,axiom,
! [A3: set_nat,N: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A3))
=> ( ~ pp(aa_set_nat_bool(member_nat(N),A3))
=> ( nat_set_encode(insert_nat(N,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))),N)),nat_set_encode(A3)) ) ) ) ).
% set_encode_insert
tff(fact_5118_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_5119_atLeast0__atMost__Suc,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).
% atLeast0_atMost_Suc
tff(fact_5120_atLeast0__lessThan__Suc,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).
% atLeast0_lessThan_Suc
tff(fact_5121_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) ) ) ).
% Icc_eq_insert_lb_nat
tff(fact_5122_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ) ).
% atLeastAtMostSuc_conv
tff(fact_5123_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) ) ) ).
% atLeastAtMost_insertL
tff(fact_5124_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_5125_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,Z)))
=> ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).
% set_decode_plus_power_2
tff(fact_5126_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
<=> ! [M5: nat] :
? [N3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N3))
& pp(aa_set_nat_bool(member_nat(N3),S3)) ) ) ).
% infinite_nat_iff_unbounded_le
tff(fact_5127_integer__of__num_I3_J,axiom,
! [N: num] : aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(code_integer_of_num,N)),aa_num_Code_integer(code_integer_of_num,N))),one_one_Code_integer) ).
% integer_of_num(3)
tff(fact_5128_card__Collect__less__nat,axiom,
! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) = N ).
% card_Collect_less_nat
tff(fact_5129_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% drop_bit_nonnegative_int_iff
tff(fact_5130_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% drop_bit_negative_int_iff
tff(fact_5131_drop__bit__minus__one,axiom,
! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% drop_bit_minus_one
tff(fact_5132_card__atLeastLessThan,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).
% card_atLeastLessThan
tff(fact_5133_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_5134_card__Collect__le__nat,axiom,
! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,N))) = aa_nat_nat(suc,N) ).
% card_Collect_le_nat
tff(fact_5135_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_5136_drop__bit__of__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% drop_bit_of_Suc_0
tff(fact_5137_card__atLeastLessThan__int,axiom,
! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).
% card_atLeastLessThan_int
tff(fact_5138_card__atLeastAtMost__int,axiom,
! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).
% card_atLeastAtMost_int
tff(fact_5139_drop__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se3928097537394005634nteger,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Xa2),X)) ).
% drop_bit_integer.abs_eq
tff(fact_5140_integer__of__num__def,axiom,
code_integer_of_num = numera6620942414471956472nteger ).
% integer_of_num_def
tff(fact_5141_card__less__Suc2,axiom,
! [M2: set_nat,I: nat] :
( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M2))
=> ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_da(M2),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_db(M2),I))) ) ) ).
% card_less_Suc2
tff(fact_5142_card__less__Suc,axiom,
! [M2: set_nat,I: nat] :
( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M2))
=> ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_da(M2),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_db(M2),I))) ) ) ).
% card_less_Suc
tff(fact_5143_card__less,axiom,
! [M2: set_nat,I: nat] :
( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M2))
=> ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_db(M2),I))) != zero_zero_nat ) ) ).
% card_less
tff(fact_5144_card__atLeastZeroLessThan__int,axiom,
! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).
% card_atLeastZeroLessThan_int
tff(fact_5145_subset__card__intvl__is__intvl,axiom,
! [A3: set_nat,K: nat] :
( pp(aa_set_nat_bool(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_5146_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
=> ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).
% atLeastAtMostPlus1_int_conv
tff(fact_5147_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set_nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(N4)),N)) ) ).
% subset_eq_atLeast0_lessThan_card
tff(fact_5148_card__sum__le__nat__sum,axiom,
! [S3: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups3542108847815614940at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S3)))),groups3542108847815614940at_nat(aTP_Lamm_bt,S3))) ).
% card_sum_le_nat_sum
tff(fact_5149_integer__of__num__triv_I1_J,axiom,
aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).
% integer_of_num_triv(1)
tff(fact_5150_integer__of__num_I2_J,axiom,
! [N: num] : aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(code_integer_of_num,N)),aa_num_Code_integer(code_integer_of_num,N)) ).
% integer_of_num(2)
tff(fact_5151_integer__of__num__triv_I2_J,axiom,
aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).
% integer_of_num_triv(2)
tff(fact_5152_pred__def,axiom,
! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bt),Nat) ).
% pred_def
tff(fact_5153_and__int_Osimps,axiom,
! [K: int,L: int] :
( ( ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
& ( ~ ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).
% and_int.simps
tff(fact_5154_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
=> ( ( ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
& ( ~ ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).
% and_int.elims
tff(fact_5155_Suc__funpow,axiom,
! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).
% Suc_funpow
tff(fact_5156_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_5157_simp__from__to,axiom,
! [J: int,I: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).
% simp_from_to
tff(fact_5158_set__decode__zero,axiom,
aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).
% set_decode_zero
tff(fact_5159_set__encode__empty,axiom,
nat_set_encode(bot_bot_set_nat) = zero_zero_nat ).
% set_encode_empty
tff(fact_5160_atLeastLessThan__singleton,axiom,
! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).
% atLeastLessThan_singleton
tff(fact_5161_atMost__0,axiom,
aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).
% atMost_0
tff(fact_5162_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
tff(fact_5163_atLeastLessThan0,axiom,
! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).
% atLeastLessThan0
tff(fact_5164_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
tff(fact_5165_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).
% atLeast1_atMost_eq_remove0
tff(fact_5166_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
tff(fact_5167_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).
% card_greaterThanLessThan_int
tff(fact_5168_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : finite_finite_int(set_or5832277885323065728an_int(L,U)) ).
% finite_greaterThanLessThan_int
tff(fact_5169_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_5170_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_5171_max__nat_Osemilattice__neutr__order__axioms,axiom,
semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bg,aTP_Lamm_bh) ).
% max_nat.semilattice_neutr_order_axioms
tff(fact_5172_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U))) ).
% finite_greaterThanLessThan
tff(fact_5173_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or4266950643985792945nteger(L,U)) ).
% finite_greaterThanLessThan_integer
tff(fact_5174_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_5175_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_5176_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
=> ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5177_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_5178_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).
% num_of_nat.simps(2)
tff(fact_5179_num__of__nat__numeral__eq,axiom,
! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).
% num_of_nat_numeral_eq
tff(fact_5180_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_5181_inc__BitM__eq,axiom,
! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).
% inc_BitM_eq
tff(fact_5182_BitM__inc__eq,axiom,
! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).
% BitM_inc_eq
tff(fact_5183_num__of__nat_Osimps_I1_J,axiom,
aa_nat_num(num_of_nat,zero_zero_nat) = one ).
% num_of_nat.simps(1)
tff(fact_5184_eval__nat__numeral_I2_J,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).
% eval_nat_numeral(2)
tff(fact_5185_BitM__plus__one,axiom,
! [N: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).
% BitM_plus_one
tff(fact_5186_one__plus__BitM,axiom,
! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).
% one_plus_BitM
tff(fact_5187_numeral__num__of__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).
% numeral_num_of_nat
tff(fact_5188_num__of__nat__One,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
=> ( aa_nat_num(num_of_nat,N) = one ) ) ).
% num_of_nat_One
tff(fact_5189_sub__BitM__One__eq,axiom,
! [N: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(N),one)) ).
% sub_BitM_One_eq
tff(fact_5190_num__of__nat__double,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).
% num_of_nat_double
tff(fact_5191_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).
% num_of_nat_plus_distrib
tff(fact_5192_or__minus__numerals_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,N)))) ).
% or_minus_numerals(5)
tff(fact_5193_or__minus__numerals_I1_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,N)))) ).
% or_minus_numerals(1)
tff(fact_5194_num__of__integer_Oabs__eq,axiom,
! [X: int] : aa_Code_integer_num(code_num_of_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_num(num_of_nat,aa_int_nat(nat2,X)) ).
% num_of_integer.abs_eq
tff(fact_5195_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
=> ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5196_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_dc(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).
% take_bit_numeral_minus_numeral_int
tff(fact_5197_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U))) ).
% finite_greaterThanAtMost
tff(fact_5198_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).
% card_greaterThanAtMost
tff(fact_5199_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_5200_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
=> ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).
% sorted_list_of_set_greaterThanAtMost
tff(fact_5201_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).
% and_minus_numerals(7)
tff(fact_5202_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).
% and_minus_numerals(3)
tff(fact_5203_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).
% and_minus_numerals(8)
tff(fact_5204_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).
% and_minus_numerals(4)
tff(fact_5205_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : finite_finite_int(set_or6656581121297822940st_int(L,U)) ).
% finite_greaterThanAtMost_int
tff(fact_5206_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or2715278749043346189nteger(L,U)) ).
% finite_greaterThanAtMost_integer
tff(fact_5207_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).
% card_greaterThanAtMost_int
tff(fact_5208_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_5209_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_5210_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_5211_int__numeral__not__and__num,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ).
% int_numeral_not_and_num
tff(fact_5212_int__numeral__and__not__num,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ).
% int_numeral_and_not_num
tff(fact_5213_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,X)),aa_Code_integer_nat(code_nat_of_integer,Y)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ) ).
% nat_of_integer_less_iff
tff(fact_5214_nat__of__integer__of__nat,axiom,
! [N: nat] : aa_Code_integer_nat(code_nat_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = N ).
% nat_of_integer_of_nat
tff(fact_5215_of__nat__of__integer,axiom,
! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_Code_integer_nat(code_nat_of_integer,K)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),K) ).
% of_nat_of_integer
tff(fact_5216_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).
% nat_of_integer_non_positive
tff(fact_5217_nat__of__integer_Oabs__eq,axiom,
! [X: int] : aa_Code_integer_nat(code_nat_of_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(nat2,X) ).
% nat_of_integer.abs_eq
tff(fact_5218_upto__rec2,axiom,
! [I: int,J: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).
% upto_rec2
tff(fact_5219_upto__split2,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
=> ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).
% upto_split2
tff(fact_5220_upto__split1,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
=> ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).
% upto_split1
tff(fact_5221_upto__split3,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
=> ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).
% upto_split3
tff(fact_5222_Gcd__remove0__nat,axiom,
! [M2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,M2))
=> ( gcd_Gcd_nat(M2) = gcd_Gcd_nat(minus_minus_set_nat(M2,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).
% Gcd_remove0_nat
tff(fact_5223_Gcd__nat__eq__one,axiom,
! [N4: set_nat] :
( pp(aa_set_nat_bool(member_nat(one_one_nat),N4))
=> ( gcd_Gcd_nat(N4) = one_one_nat ) ) ).
% Gcd_nat_eq_one
tff(fact_5224_Gcd__dvd__nat,axiom,
! [A: nat,A3: set_nat] :
( pp(aa_set_nat_bool(member_nat(A),A3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,gcd_Gcd_nat(A3)),A)) ) ).
% Gcd_dvd_nat
tff(fact_5225_Gcd__greatest__nat,axiom,
! [A3: set_nat,A: nat] :
( ! [B2: nat] :
( pp(aa_set_nat_bool(member_nat(B2),A3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B2)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),gcd_Gcd_nat(A3))) ) ).
% Gcd_greatest_nat
tff(fact_5226_abs__Gcd__eq,axiom,
! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Gcd_int(K5)) = gcd_Gcd_int(K5) ).
% abs_Gcd_eq
tff(fact_5227_finite__lessThan,axiom,
! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K))) ).
% finite_lessThan
tff(fact_5228_card__lessThan,axiom,
! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).
% card_lessThan
tff(fact_5229_lessThan__0,axiom,
aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).
% lessThan_0
tff(fact_5230_Gcd__dvd__int,axiom,
! [A: int,A3: set_int] :
( member_int(A,A3)
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,gcd_Gcd_int(A3)),A)) ) ).
% Gcd_dvd_int
tff(fact_5231_Gcd__greatest__int,axiom,
! [A3: set_int,A: int] :
( ! [B2: int] :
( member_int(B2,A3)
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B2)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),gcd_Gcd_int(A3))) ) ).
% Gcd_greatest_int
tff(fact_5232_lessThan__atLeast0,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ).
% lessThan_atLeast0
tff(fact_5233_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_5234_lessThan__empty__iff,axiom,
! [N: nat] :
( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
<=> ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
tff(fact_5235_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_5236_Gcd__int__greater__eq__0,axiom,
! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5))) ).
% Gcd_int_greater_eq_0
tff(fact_5237_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_5238_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).
% atLeast1_lessThan_eq_remove0
tff(fact_5239_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
=> ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).
% image_atLeastZeroLessThan_integer
tff(fact_5240_take__bit__num__simps_I7_J,axiom,
! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R),M))) ).
% take_bit_num_simps(7)
tff(fact_5241_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).
% take_bit_num_simps(4)
tff(fact_5242_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_dd,bit_and_not_num(M,N)) ).
% and_not_num.simps(8)
tff(fact_5243_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_5244_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_5245_Gcd__abs__eq,axiom,
! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).
% Gcd_abs_eq
tff(fact_5246_Gcd__int__eq,axiom,
! [N4: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N4)) = aa_nat_int(semiri1314217659103216013at_int,gcd_Gcd_nat(N4)) ).
% Gcd_int_eq
tff(fact_5247_Gcd__nat__abs__eq,axiom,
! [K5: set_int] : gcd_Gcd_nat(image_int_nat(aTP_Lamm_de,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).
% Gcd_nat_abs_eq
tff(fact_5248_zero__notin__Suc__image,axiom,
! [A3: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A3))) ).
% zero_notin_Suc_image
tff(fact_5249_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% image_int_atLeastAtMost
tff(fact_5250_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% image_int_atLeastLessThan
tff(fact_5251_image__Suc__atMost,axiom,
! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).
% image_Suc_atMost
tff(fact_5252_image__Suc__lessThan,axiom,
! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).
% image_Suc_lessThan
tff(fact_5253_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).
% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5254_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).
% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5255_atMost__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% atMost_Suc_eq_insert_0
tff(fact_5256_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% lessThan_Suc_eq_insert_0
tff(fact_5257_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] : image_int_int(aTP_Lamm_df(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).
% image_add_int_atLeastLessThan
tff(fact_5258_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_dg(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,U),L))) = set_or8404916559141939852nteger(L,U) ).
% image_add_integer_atLeastLessThan
tff(fact_5259_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
=> ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).
% image_atLeastZeroLessThan_int
tff(fact_5260_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y: nat,X: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_5261_take__bit__num__simps_I6_J,axiom,
! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_di,bit_take_bit_num(aa_num_nat(pred_numeral,R),M)) ).
% take_bit_num_simps(6)
tff(fact_5262_take__bit__num__simps_I1_J,axiom,
! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).
% take_bit_num_simps(1)
tff(fact_5263_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_di,bit_take_bit_num(N,M)) ).
% take_bit_num_simps(3)
tff(fact_5264_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( bit_and_not_num(M,N) = none_num )
<=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).
% and_not_num_eq_None_iff
tff(fact_5265_take__bit__num__def,axiom,
! [N: nat,M: num] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
=> ( bit_take_bit_num(N,M) = none_num ) )
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
=> ( bit_take_bit_num(N,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).
% take_bit_num_def
tff(fact_5266_and__not__num_Oelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( bit_and_not_num(X,Xa2) = Y )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit0,N2)
=> ( Y != some_num(one) ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit1,N2)
=> ( Y != none_num ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_dd,bit_and_not_num(M3,N2)) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
tff(fact_5267_Gcd__int__def,axiom,
! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,gcd_Gcd_nat(image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).
% Gcd_int_def
tff(fact_5268_and__num_Oelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y != some_num(one) ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit0,N2)
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit1,N2)
=> ( Y != some_num(one) ) ) )
=> ( ( ? [M3: num] : X = aa_num_num(bit0,M3)
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N2)) ) ) )
=> ( ( ? [M3: num] : X = aa_num_num(bit1,M3)
=> ( ( Xa2 = one )
=> ( Y != some_num(one) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N2)) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_dd,bit_un7362597486090784418nd_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
tff(fact_5269_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] : bit_un7362597486090784418nd_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_dd,bit_un7362597486090784418nd_num(M,N)) ).
% and_num.simps(9)
tff(fact_5270_Code__Numeral_Onegative__def,axiom,
code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).
% Code_Numeral.negative_def
tff(fact_5271_measure__function__int,axiom,
fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).
% measure_function_int
tff(fact_5272_nat__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Code_integer_nat(code_nat_of_integer,K) = produc1555791787009142072er_nat(aTP_Lamm_dj,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).
% nat_of_integer_code
tff(fact_5273_num__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
=> ( aa_Code_integer_num(code_num_of_integer,K) = one ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
=> ( aa_Code_integer_num(code_num_of_integer,K) = produc7336495610019696514er_num(aTP_Lamm_dk,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).
% num_of_integer_code
tff(fact_5274_int__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Code_integer_int(code_int_of_integer,K) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( ( ( K = zero_z3403309356797280102nteger )
=> ( aa_Code_integer_int(code_int_of_integer,K) = zero_zero_int ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_int(code_int_of_integer,K) = produc1553301316500091796er_int(aTP_Lamm_dl,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ) ) ).
% int_of_integer_code
tff(fact_5275_int__of__integer__inverse,axiom,
! [X: code_integer] : aa_int_Code_integer(code_integer_of_int,aa_Code_integer_int(code_int_of_integer,X)) = X ).
% int_of_integer_inverse
tff(fact_5276_int__of__integer__integer__of__int,axiom,
! [K: int] : aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,K)) = K ).
% int_of_integer_integer_of_int
tff(fact_5277_integer__of__int__int__of__integer,axiom,
! [K: code_integer] : aa_int_Code_integer(code_integer_of_int,aa_Code_integer_int(code_int_of_integer,K)) = K ).
% integer_of_int_int_of_integer
tff(fact_5278_int__of__integer__sub,axiom,
! [K: num,L: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(neg_nu5755505904847501662nteger(K),L)) = aa_num_int(neg_numeral_sub_int(K),L) ).
% int_of_integer_sub
tff(fact_5279_of__int__integer__of,axiom,
! [K: code_integer] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_Code_integer_int(code_int_of_integer,K)) = K ).
% of_int_integer_of
tff(fact_5280_int__of__integer__of__int,axiom,
! [K: int] : aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(ring_18347121197199848620nteger,K)) = K ).
% int_of_integer_of_int
tff(fact_5281_zero__integer_Orep__eq,axiom,
aa_Code_integer_int(code_int_of_integer,zero_z3403309356797280102nteger) = zero_zero_int ).
% zero_integer.rep_eq
tff(fact_5282_int__of__integer__numeral,axiom,
! [K: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).
% int_of_integer_numeral
tff(fact_5283_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% plus_integer.rep_eq
tff(fact_5284_one__integer_Orep__eq,axiom,
aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).
% one_integer.rep_eq
tff(fact_5285_uminus__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% uminus_integer.rep_eq
tff(fact_5286_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% times_integer.rep_eq
tff(fact_5287_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% minus_integer.rep_eq
tff(fact_5288_abs__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)) = aa_int_int(abs_abs_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% abs_integer.rep_eq
tff(fact_5289_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% divide_integer.rep_eq
tff(fact_5290_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% modulo_integer.rep_eq
tff(fact_5291_int__of__integer__of__nat,axiom,
! [N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% int_of_integer_of_nat
tff(fact_5292_sgn__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)) = aa_int_int(sgn_sgn_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% sgn_integer.rep_eq
tff(fact_5293_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(ord_max_Code_integer(K),L)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ).
% int_of_integer_max
tff(fact_5294_integer__eq__iff,axiom,
! [K: code_integer,L: code_integer] :
( ( K = L )
<=> ( aa_Code_integer_int(code_int_of_integer,K) = aa_Code_integer_int(code_int_of_integer,L) ) ) ).
% integer_eq_iff
tff(fact_5295_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( aa_Code_integer_int(code_int_of_integer,K) = aa_Code_integer_int(code_int_of_integer,L) )
=> ( K = L ) ) ).
% integer_eqI
tff(fact_5296_int__of__integer__inject,axiom,
! [X: code_integer,Y: code_integer] :
( ( aa_Code_integer_int(code_int_of_integer,X) = aa_Code_integer_int(code_int_of_integer,Y) )
<=> ( X = Y ) ) ).
% int_of_integer_inject
tff(fact_5297_less__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa2))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2))) ) ).
% less_integer.rep_eq
tff(fact_5298_integer__less__iff,axiom,
! [K: code_integer,L: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L))) ) ).
% integer_less_iff
tff(fact_5299_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Y)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).
% int_of_integer_less_iff
tff(fact_5300_less__eq__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa2))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2))) ) ).
% less_eq_integer.rep_eq
tff(fact_5301_integer__less__eq__iff,axiom,
! [K: code_integer,L: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L))) ) ).
% integer_less_eq_iff
tff(fact_5302_nat__of__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_nat(code_nat_of_integer,X) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).
% nat_of_integer.rep_eq
tff(fact_5303_take__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,X),Xa2)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% take_bit_integer.rep_eq
tff(fact_5304_not__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% not_integer.rep_eq
tff(fact_5305_and__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% and_integer.rep_eq
tff(fact_5306_or__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% or_integer.rep_eq
tff(fact_5307_bit__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Cod2498268374540730821t_bool(bit_se9216721137139052372nteger,X) = aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% bit_integer.rep_eq
tff(fact_5308_xor__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% xor_integer.rep_eq
tff(fact_5309_push__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se7788150548672797655nteger,X),Xa2)) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% push_bit_integer.rep_eq
tff(fact_5310_drop__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se3928097537394005634nteger,X),Xa2)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% drop_bit_integer.rep_eq
tff(fact_5311_mask__integer_Orep__eq,axiom,
! [X: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(bit_se2119862282449309892nteger,X)) = aa_nat_int(bit_se2000444600071755411sk_int,X) ).
% mask_integer.rep_eq
tff(fact_5312_unset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,X),Xa2)) = aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% unset_bit_integer.rep_eq
tff(fact_5313_set__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,X),Xa2)) = aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% set_bit_integer.rep_eq
tff(fact_5314_flip__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,X),Xa2)) = aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% flip_bit_integer.rep_eq
tff(fact_5315_num__of__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_num(code_num_of_integer,X) = aa_nat_num(num_of_nat,aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X))) ).
% num_of_integer.rep_eq
tff(fact_5316_int__ge__less__than__def,axiom,
! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_dm(D))) ).
% int_ge_less_than_def
tff(fact_5317_int__ge__less__than2__def,axiom,
! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_dn(D))) ).
% int_ge_less_than2_def
tff(fact_5318_Divides_Oadjust__div__def,axiom,
! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_do,Qr) ).
% Divides.adjust_div_def
tff(fact_5319_divmod__step__integer__def,axiom,
! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_dp(L),Qr) ).
% divmod_step_integer_def
tff(fact_5320_divmod__integer__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,K),L)) ).
% divmod_integer_def
tff(fact_5321_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_bool_int(zero_n2684676970156552555ol_int,fNot(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)))) ).
% Divides.adjust_div_eq
tff(fact_5322_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dq(L)),Qr) ).
% divmod_step_nat_def
tff(fact_5323_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_dr(L),Qr) ).
% divmod_step_int_def
tff(fact_5324_take__bit__num__code,axiom,
! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_dv,product_Pair_nat_num(N,M)) ).
% take_bit_num_code
tff(fact_5325_minus__one__div__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).
% minus_one_div_numeral
tff(fact_5326_one__div__minus__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).
% one_div_minus_numeral
tff(fact_5327_divmod__BitM__2__eq,axiom,
! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),one_one_int) ).
% divmod_BitM_2_eq
tff(fact_5328_divmod__integer_H__def,axiom,
! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).
% divmod_integer'_def
tff(fact_5329_divmod__nat__if,axiom,
! [N: nat,M: nat] :
( ( ( ( N = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
=> ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
& ( ~ ( ( N = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
=> ( divmod_nat(M,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dw),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).
% divmod_nat_if
tff(fact_5330_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)))
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_5331_eucl__rel__int__by0,axiom,
! [K: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K))) ).
% eucl_rel_int_by0
tff(fact_5332_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q2: int] :
( ( L != zero_zero_int )
=> ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),zero_zero_int))) ) ) ).
% eucl_rel_int_dividesI
tff(fact_5333_zminus1__lemma,axiom,
! [A: int,B: int,Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)))
=> ( ( B != zero_zero_int )
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),aa_int7462565939943104719nt_int(product_Pair_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int),aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R))))) ) ) ).
% zminus1_lemma
tff(fact_5334_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)))
<=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),L)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_5335_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q2: int] :
( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,L) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,L)))
=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R) )
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R))) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_5336_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
<=> ( ? [K2: int] :
( ( A1 = K2 )
& ( A22 = zero_zero_int )
& ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K2) ) )
| ? [L3: int,K2: int,Q6: int] :
( ( A1 = K2 )
& ( A22 = L3 )
& ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),zero_zero_int) )
& ( L3 != zero_zero_int )
& ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3) ) )
| ? [R4: int,L3: int,K2: int,Q6: int] :
( ( A1 = K2 )
& ( A22 = L3 )
& ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R4) )
& ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,L3) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R4)),aa_int_int(abs_abs_int,L3)))
& ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3)),R4) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_5337_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
=> ( ( ( A22 = zero_zero_int )
=> ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
=> ( ! [Q4: int] :
( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int) )
=> ( ( A22 != zero_zero_int )
=> ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22) ) ) )
=> ~ ! [R5: int,Q4: int] :
( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R5) )
=> ( ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,A22) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),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,Q4),A22)),R5) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_5338_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),R)))
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R))))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_5339_and__int_Opsimps,axiom,
! [K: int,L: int] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K),L)))
=> ( ( ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
& ( ~ ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).
% and_int.psimps
tff(fact_5340_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
=> ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2)))
=> ~ ( ( ( ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
& ( ~ ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,fconj(fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),fNot(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
=> ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2))) ) ) ) ).
% and_int.pelims
tff(fact_5341_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_bool] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1)))
=> ( ! [K3: int,L2: int] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K3),L2)))
=> ( ( ~ ( member_int(K3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
& member_int(L2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,K3),L2)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).
% and_int.pinduct
tff(fact_5342_upto_Opelims,axiom,
! [X: int,Xa2: int,Y: list_int] :
( ( upto(X,Xa2) = Y )
=> ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2)))
=> ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
=> ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
=> ( Y = nil_int ) ) )
=> ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2))) ) ) ) ).
% upto.pelims
tff(fact_5343_upto_Opsimps,axiom,
! [I: int,J: int] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I),J)))
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(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)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( upto(I,J) = nil_int ) ) ) ) ).
% upto.psimps
tff(fact_5344_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_bool] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1)))
=> ( ! [I2: int,J2: int] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J2)))
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
=> pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J2)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).
% upto.pinduct
tff(fact_5345_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( ( ( L = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
& ( ( L != zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_dx(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).
% divmod_integer_eq_cases
tff(fact_5346_divmod__abs__code_I6_J,axiom,
! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).
% divmod_abs_code(6)
tff(fact_5347_divmod__abs__code_I5_J,axiom,
! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).
% divmod_abs_code(5)
tff(fact_5348_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
=> ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
=> ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_dy(L),code_divmod_abs(K,L)) ) ) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
=> ( ( ( L = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
& ( ( L != zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_dz(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).
% divmod_integer_code
tff(fact_5349_divmod__abs__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ).
% divmod_abs_def
tff(fact_5350_bit__cut__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_ea(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).
% bit_cut_integer_code
tff(fact_5351_normalize__negative,axiom,
! [Q2: int,P3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q2),zero_zero_int))
=> ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3)),aa_int_int(uminus_uminus_int,Q2))) ) ) ).
% normalize_negative
tff(fact_5352_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = produc5135875973608410785r_bool(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),fNot(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K))) ).
% bit_cut_integer_def
tff(fact_5353_normalize__denom__zero,axiom,
! [P3: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).
% normalize_denom_zero
tff(fact_5354_normalize__denom__pos,axiom,
! [R: product_prod_int_int,P3: int,Q2: int] :
( ( normalize(R) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).
% normalize_denom_pos
tff(fact_5355_normalize__crossproduct,axiom,
! [Q2: int,S: int,P3: int,R: int] :
( ( Q2 != zero_zero_int )
=> ( ( S != zero_zero_int )
=> ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R),S)) )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R),Q2) ) ) ) ) ).
% normalize_crossproduct
tff(fact_5356_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).
% prod_decode_aux.elims
tff(fact_5357_prod__decode__aux_Osimps,axiom,
! [M: nat,K: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
=> ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
=> ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).
% prod_decode_aux.simps
tff(fact_5358_bezw__0,axiom,
! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ).
% bezw_0
tff(fact_5359_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ec),Xa2),X)) ).
% times_int.abs_eq
tff(fact_5360_int_Oabs__induct,axiom,
! [P: fun_int_bool,X: int] :
( ! [Y2: product_prod_nat_nat] : pp(aa_int_bool(P,aa_Pro4240683299535347697at_int(abs_Integ,Y2)))
=> pp(aa_int_bool(P,X)) ) ).
% int.abs_induct
tff(fact_5361_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X2: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) ).
% eq_Abs_Integ
tff(fact_5362_nat_Oabs__eq,axiom,
! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).
% nat.abs_eq
tff(fact_5363_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_5364_int__def,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).
% int_def
tff(fact_5365_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ed),X)) ).
% uminus_int.abs_eq
tff(fact_5366_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_5367_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_ef),Xa2),X)) ) ).
% less_int.abs_eq
tff(fact_5368_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_eh),Xa2),X)) ) ).
% less_eq_int.abs_eq
tff(fact_5369_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ej),Xa2),X)) ).
% plus_int.abs_eq
tff(fact_5370_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_el),Xa2),X)) ).
% minus_int.abs_eq
tff(fact_5371_rat__minus__code,axiom,
! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_en(Q2),quotient_of(P3)) ).
% rat_minus_code
tff(fact_5372_rat__plus__code,axiom,
! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_ep(Q2),quotient_of(P3)) ).
% rat_plus_code
tff(fact_5373_sort__upto,axiom,
! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_bu,upto(I,J)) = upto(I,J) ).
% sort_upto
tff(fact_5374_rat__one__code,axiom,
quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int) ).
% rat_one_code
tff(fact_5375_rat__zero__code,axiom,
quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).
% rat_zero_code
tff(fact_5376_quotient__of__number_I3_J,axiom,
! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int) ).
% quotient_of_number(3)
tff(fact_5377_quotient__of__number_I4_J,axiom,
quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).
% quotient_of_number(4)
tff(fact_5378_quotient__of__number_I5_J,axiom,
! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).
% quotient_of_number(5)
tff(fact_5379_diff__rat__def,axiom,
! [Q2: rat,R: rat] : aa_rat_rat(minus_minus_rat(Q2),R) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q2),aa_rat_rat(uminus_uminus_rat,R)) ).
% diff_rat_def
tff(fact_5380_quotient__of__denom__pos,axiom,
! [R: rat,P3: int,Q2: int] :
( ( quotient_of(R) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).
% quotient_of_denom_pos
tff(fact_5381_rat__floor__code,axiom,
! [P3: rat] : aa_rat_int(archim3151403230148437115or_rat,P3) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P3)) ).
% rat_floor_code
tff(fact_5382_rat__uminus__code,axiom,
! [P3: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_eq,quotient_of(P3)) ).
% rat_uminus_code
tff(fact_5383_rat__abs__code,axiom,
! [P3: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_er,quotient_of(P3)) ).
% rat_abs_code
tff(fact_5384_rat__less__eq__code,axiom,
! [P3: rat,Q2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),Q2))
<=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_et(Q2)),quotient_of(P3))) ) ).
% rat_less_eq_code
tff(fact_5385_rat__less__code,axiom,
! [P3: rat,Q2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q2))
<=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_ev(Q2)),quotient_of(P3))) ) ).
% rat_less_code
tff(fact_5386_rat__divide__code,axiom,
! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_ex(Q2),quotient_of(P3)) ).
% rat_divide_code
tff(fact_5387_rat__times__code,axiom,
! [P3: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_ez(Q2),quotient_of(P3)) ).
% rat_times_code
tff(fact_5388_quotient__of__int,axiom,
! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(product_Pair_int_int(A),one_one_int) ).
% quotient_of_int
tff(fact_5389_rat__inverse__code,axiom,
! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_fa,quotient_of(P3)) ).
% rat_inverse_code
tff(fact_5390_sgn__rat__def,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).
% sgn_rat_def
tff(fact_5391_Frct__code__post_I5_J,axiom,
! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).
% Frct_code_post(5)
tff(fact_5392_pow_Osimps_I1_J,axiom,
! [X: num] : aa_num_num(pow(X),one) = X ).
% pow.simps(1)
tff(fact_5393_Frct__code__post_I1_J,axiom,
! [A: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A)) = zero_zero_rat ).
% Frct_code_post(1)
tff(fact_5394_Frct__code__post_I2_J,axiom,
! [A: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),zero_zero_int)) = zero_zero_rat ).
% Frct_code_post(2)
tff(fact_5395_Frct__code__post_I3_J,axiom,
frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) = one_one_rat ).
% Frct_code_post(3)
tff(fact_5396_Frct__code__post_I4_J,axiom,
! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).
% Frct_code_post(4)
tff(fact_5397_less__eq__int_Orep__eq,axiom,
! [X: int,Xa2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_eh),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).
% less_eq_int.rep_eq
tff(fact_5398_nat_Orep__eq,axiom,
! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).
% nat.rep_eq
tff(fact_5399_less__int_Orep__eq,axiom,
! [X: int,Xa2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_ef),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).
% less_int.rep_eq
tff(fact_5400_sort__upt,axiom,
! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_bt,upt(M,N)) = upt(M,N) ).
% sort_upt
tff(fact_5401_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( upt(zero_zero_nat,J) = nil_nat )
<=> ( J = zero_zero_nat ) ) ).
% upt_0_eq_Nil_conv
tff(fact_5402_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K)) )
=> ( append_nat(upt(I,J),upt(J,K)) = upt(I,K) ) ) ).
% upt_merge
tff(fact_5403_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( upt(I,J) = nil_nat ) ) ).
% upt_conv_Nil
tff(fact_5404_drop__upt,axiom,
! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).
% drop_upt
tff(fact_5405_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( upt(I,J) = nil_nat )
<=> ( ( J = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).
% upt_eq_Nil_conv
tff(fact_5406_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
=> ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).
% take_upt
tff(fact_5407_length__upt,axiom,
! [I: nat,J: nat] : aa_list_nat_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_5408_last__upt,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5409_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J))
=> ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).
% nth_upt
tff(fact_5410_map__add__upt_H,axiom,
! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).
% map_add_upt'
tff(fact_5411_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( ( upt(I,J) = append_nat(Xs,Ys) )
<=> ? [K2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
& ( upt(I,K2) = Xs )
& ( upt(K2,J) = Ys ) ) ) ) ).
% upt_eq_append_conv
tff(fact_5412_upt__0,axiom,
! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).
% upt_0
tff(fact_5413_upt__append,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).
% upt_append
tff(fact_5414_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).
% upt_add_eq_append
tff(fact_5415_butlast__upt,axiom,
! [M: nat,N: nat] : butlast_nat(upt(M,N)) = upt(M,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).
% butlast_upt
tff(fact_5416_atLeast__upt,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).
% atLeast_upt
tff(fact_5417_map__add__upt,axiom,
! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
% map_add_upt
tff(fact_5418_sorted__upt,axiom,
! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).
% sorted_upt
tff(fact_5419_atMost__upto,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).
% atMost_upto
tff(fact_5420_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).
% upt_Suc_append
tff(fact_5421_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).
% upt_Suc
tff(fact_5422_upt__filter__extend,axiom,
! [U: nat,U2: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),U2))
=> ( ! [I2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),I2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),U2)) )
=> ~ pp(aa_nat_bool(P,I2)) )
=> ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U2)) ) ) ) ).
% upt_filter_extend
tff(fact_5423_map__decr__upt,axiom,
! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_fc,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).
% map_decr_upt
tff(fact_5424_upt__eq__lel__conv,axiom,
! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
( ( upt(L,H) = append_nat(Is1,cons_nat(I,Is2)) )
<=> ( ( Is1 = upt(L,I) )
& ( Is2 = upt(aa_nat_nat(suc,I),H) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),H)) ) ) ).
% upt_eq_lel_conv
tff(fact_5425_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( upt(I,J) = cons_nat(X,Xs) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& ( I = X )
& ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_5426_sum__list__upt,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups4561878855575611511st_nat(upt(M,N)) = groups3542108847815614940at_nat(aTP_Lamm_bt,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).
% sum_list_upt
tff(fact_5427_card__length__sum__list__rec,axiom,
! [M: nat,N4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
=> ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fd(M),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fe(M),N4)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ff(M),N4)))) ) ) ).
% card_length_sum_list_rec
tff(fact_5428_card__length__sum__list,axiom,
! [M: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fd(M),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),M)),one_one_nat)),N4) ).
% card_length_sum_list
tff(fact_5429_floor__rat__def,axiom,
! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_fg(X)) ).
% floor_rat_def
tff(fact_5430_uminus__int__def,axiom,
uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_ed)) ).
% uminus_int_def
tff(fact_5431_times__int__def,axiom,
times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ec)) ).
% times_int_def
tff(fact_5432_minus__int__def,axiom,
minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_el)) ).
% minus_int_def
tff(fact_5433_plus__int__def,axiom,
plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ej)) ).
% plus_int_def
tff(fact_5434_prod__encode__def,axiom,
nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_fh) ).
% prod_encode_def
tff(fact_5435_le__prod__encode__1,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).
% le_prod_encode_1
tff(fact_5436_le__prod__encode__2,axiom,
! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).
% le_prod_encode_2
tff(fact_5437_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).
% prod_encode_prod_decode_aux
tff(fact_5438_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X = cons_nat(X2,Xs2) )
=> ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) ) ) ) ) ).
% list_encode.elims
tff(fact_5439_list__encode_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X = cons_nat(X2,Xs2) )
=> ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) )
=> ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X2,Xs2))) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_5440_list__encode_Osimps_I1_J,axiom,
nat_list_encode(nil_nat) = zero_zero_nat ).
% list_encode.simps(1)
tff(fact_5441_rat__floor__lemma,axiom,
! [A: int,B: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)))) ) ).
% rat_floor_lemma
tff(fact_5442_less__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).
% less_rat
tff(fact_5443_le__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_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_5444_add__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_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_5445_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(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_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_5446_eq__rat_I3_J,axiom,
! [A: int,C: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),A) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),C) ).
% eq_rat(3)
tff(fact_5447_Rat__induct__pos,axiom,
! [P: fun_rat_bool,Q2: rat] :
( ! [A2: int,B2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
=> pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A2),B2))) )
=> pp(aa_rat_bool(P,Q2)) ) ).
% Rat_induct_pos
tff(fact_5448_eq__rat_I2_J,axiom,
! [A: int] : aa_int_rat(aa_int_fun_int_rat(fract,A),zero_zero_int) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).
% eq_rat(2)
tff(fact_5449_mult__rat__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_rat(aa_int_fun_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(aa_int_fun_int_rat(fract,A),B) ) ) ).
% mult_rat_cancel
tff(fact_5450_eq__rat_I1_J,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( aa_int_rat(aa_int_fun_int_rat(fract,A),B) = aa_int_rat(aa_int_fun_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_5451_Fract__of__nat__eq,axiom,
! [K: nat] : aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) = aa_nat_rat(semiri681578069525770553at_rat,K) ).
% Fract_of_nat_eq
tff(fact_5452_rat__number__collapse_I1_J,axiom,
! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),K) = zero_zero_rat ).
% rat_number_collapse(1)
tff(fact_5453_rat__number__collapse_I6_J,axiom,
! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),zero_zero_int) = zero_zero_rat ).
% rat_number_collapse(6)
tff(fact_5454_One__rat__def,axiom,
one_one_rat = aa_int_rat(aa_int_fun_int_rat(fract,one_one_int),one_one_int) ).
% One_rat_def
tff(fact_5455_Fract__of__int__eq,axiom,
! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),one_one_int) = aa_int_rat(ring_1_of_int_rat,K) ).
% Fract_of_int_eq
tff(fact_5456_Zero__rat__def,axiom,
zero_zero_rat = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).
% Zero_rat_def
tff(fact_5457_rat__number__expand_I3_J,axiom,
! [K: num] : aa_num_rat(numeral_numeral_rat,K) = aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,K)),one_one_int) ).
% rat_number_expand(3)
tff(fact_5458_rat__number__collapse_I3_J,axiom,
! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,W)),one_one_int) = aa_num_rat(numeral_numeral_rat,W) ).
% rat_number_collapse(3)
tff(fact_5459_zero__less__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).
% zero_less_Fract_iff
tff(fact_5460_Fract__less__zero__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% Fract_less_zero_iff
tff(fact_5461_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% Fract_less_one_iff
tff(fact_5462_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).
% one_less_Fract_iff
tff(fact_5463_rat__number__collapse_I5_J,axiom,
aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% rat_number_collapse(5)
tff(fact_5464_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)),N) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,M),N)),one_one_rat) ) ) ).
% Fract_add_one
tff(fact_5465_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).
% Fract_le_zero_iff
tff(fact_5466_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
% zero_le_Fract_iff
tff(fact_5467_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% Fract_le_one_iff
tff(fact_5468_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).
% one_le_Fract_iff
tff(fact_5469_rat__number__expand_I5_J,axiom,
! [K: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).
% rat_number_expand(5)
tff(fact_5470_rat__number__collapse_I4_J,axiom,
! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) ).
% rat_number_collapse(4)
tff(fact_5471_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] : aa_num_num(pow(X),aa_num_num(bit1,Y)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(X),Y))),X) ).
% pow.simps(3)
tff(fact_5472_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
tff(fact_5473_sqr_Osimps_I2_J,axiom,
! [N: num] : aa_num_num(sqr,aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,N))) ).
% sqr.simps(2)
tff(fact_5474_sqr_Osimps_I1_J,axiom,
aa_num_num(sqr,one) = one ).
% sqr.simps(1)
tff(fact_5475_less__int__def,axiom,
ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_ef)) ).
% less_int_def
tff(fact_5476_less__eq__int__def,axiom,
ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_eh)) ).
% less_eq_int_def
tff(fact_5477_sqr__conv__mult,axiom,
! [X: num] : aa_num_num(sqr,X) = aa_num_num(times_times_num(X),X) ).
% sqr_conv_mult
tff(fact_5478_nat__def,axiom,
nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).
% nat_def
tff(fact_5479_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] : aa_num_num(pow(X),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(pow(X),Y)) ).
% pow.simps(2)
tff(fact_5480_sqr_Osimps_I3_J,axiom,
! [N: num] : aa_num_num(sqr,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(sqr,N)),N))) ).
% sqr.simps(3)
tff(fact_5481_positive__rat,axiom,
! [A: int,B: int] :
( pp(aa_rat_bool(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).
% positive_rat
tff(fact_5482_less__eq__integer__def,axiom,
ord_le3102999989581377725nteger = map_fu1404520703793252016r_bool(code_int_of_integer,map_fu6396621194128205665l_bool(code_int_of_integer,id_bool),ord_less_eq_int) ).
% less_eq_integer_def
tff(fact_5483_less__integer__def,axiom,
ord_le6747313008572928689nteger = map_fu1404520703793252016r_bool(code_int_of_integer,map_fu6396621194128205665l_bool(code_int_of_integer,id_bool),ord_less_int) ).
% less_integer_def
tff(fact_5484_plus__integer__def,axiom,
plus_p5714425477246183910nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),plus_plus_int) ).
% plus_integer_def
tff(fact_5485_times__integer__def,axiom,
times_3573771949741848930nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),times_times_int) ).
% times_integer_def
tff(fact_5486_minus__integer__def,axiom,
minus_8373710615458151222nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),minus_minus_int) ).
% minus_integer_def
tff(fact_5487_divide__integer__def,axiom,
divide6298287555418463151nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),divide_divide_int) ).
% divide_integer_def
tff(fact_5488_modulo__integer__def,axiom,
modulo364778990260209775nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),modulo_modulo_int) ).
% modulo_integer_def
tff(fact_5489_positive__add,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(positive,X))
=> ( pp(aa_rat_bool(positive,Y))
=> pp(aa_rat_bool(positive,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))) ) ) ).
% positive_add
tff(fact_5490_and__integer__def,axiom,
bit_se3949692690581998587nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se725231765392027082nd_int) ).
% and_integer_def
tff(fact_5491_or__integer__def,axiom,
bit_se1080825931792720795nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se1409905431419307370or_int) ).
% or_integer_def
tff(fact_5492_take__bit__integer__def,axiom,
bit_se1745604003318907178nteger = map_fu214667362969451334nteger(id_nat,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se2923211474154528505it_int) ).
% take_bit_integer_def
tff(fact_5493_xor__integer__def,axiom,
bit_se3222712562003087583nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se6526347334894502574or_int) ).
% xor_integer_def
tff(fact_5494_drop__bit__integer__def,axiom,
bit_se3928097537394005634nteger = map_fu214667362969451334nteger(id_nat,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se8568078237143864401it_int) ).
% drop_bit_integer_def
tff(fact_5495_push__bit__integer__def,axiom,
bit_se7788150548672797655nteger = map_fu214667362969451334nteger(id_nat,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se545348938243370406it_int) ).
% push_bit_integer_def
tff(fact_5496_unset__bit__integer__def,axiom,
bit_se8260200283734997820nteger = map_fu214667362969451334nteger(id_nat,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se4203085406695923979it_int) ).
% unset_bit_integer_def
tff(fact_5497_set__bit__integer__def,axiom,
bit_se2793503036327961859nteger = map_fu214667362969451334nteger(id_nat,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se7879613467334960850it_int) ).
% set_bit_integer_def
tff(fact_5498_flip__bit__integer__def,axiom,
bit_se1345352211410354436nteger = map_fu214667362969451334nteger(id_nat,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se2159334234014336723it_int) ).
% flip_bit_integer_def
tff(fact_5499_nat__of__integer__def,axiom,
code_nat_of_integer = map_fu6539832666145259331at_nat(code_int_of_integer,id_nat,nat2) ).
% nat_of_integer_def
tff(fact_5500_bit__integer__def,axiom,
bit_se9216721137139052372nteger = map_fu7981381284939422625t_bool(code_int_of_integer,id_fun_nat_bool,bit_se1146084159140164899it_int) ).
% bit_integer_def
tff(fact_5501_uminus__integer__def,axiom,
uminus1351360451143612070nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),uminus_uminus_int) ).
% uminus_integer_def
tff(fact_5502_abs__integer__def,axiom,
abs_abs_Code_integer = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),abs_abs_int) ).
% abs_integer_def
tff(fact_5503_mask__integer__def,axiom,
bit_se2119862282449309892nteger = map_fu6290471996055670595nteger(id_nat,code_integer_of_int,bit_se2000444600071755411sk_int) ).
% mask_integer_def
tff(fact_5504_sgn__integer__def,axiom,
sgn_sgn_Code_integer = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),sgn_sgn_int) ).
% sgn_integer_def
tff(fact_5505_not__integer__def,axiom,
bit_ri7632146776885996613nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_ri7919022796975470100ot_int) ).
% not_integer_def
tff(fact_5506_num__of__integer__def,axiom,
code_num_of_integer = map_fu1227494855608507351um_num(code_int_of_integer,id_num,comp_nat_num_int(num_of_nat,nat2)) ).
% num_of_integer_def
tff(fact_5507_min__0R,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).
% min_0R
tff(fact_5508_min__0L,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).
% min_0L
tff(fact_5509_min__Suc__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).
% min_Suc_Suc
tff(fact_5510_min__Suc__gt_I2_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).
% min_Suc_gt(2)
tff(fact_5511_min__Suc__gt_I1_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).
% min_Suc_gt(1)
tff(fact_5512_min__numeral__Suc,axiom,
! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).
% min_numeral_Suc
tff(fact_5513_min__Suc__numeral,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).
% min_Suc_numeral
tff(fact_5514_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).
% nat_mult_min_left
tff(fact_5515_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).
% nat_mult_min_right
tff(fact_5516_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
tff(fact_5517_min__diff,axiom,
! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).
% min_diff
tff(fact_5518_min__Suc1,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_fi,N)),M) ).
% min_Suc1
tff(fact_5519_min__Suc2,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_fj,N)),M) ).
% min_Suc2
tff(fact_5520_integer__of__nat__def,axiom,
code_integer_of_nat = map_fu6290471996055670595nteger(id_nat,code_integer_of_int,semiri1314217659103216013at_int) ).
% integer_of_nat_def
tff(fact_5521_nat__of__integer__integer__of__nat,axiom,
! [N: nat] : aa_Code_integer_nat(code_nat_of_integer,aa_nat_Code_integer(code_integer_of_nat,N)) = N ).
% nat_of_integer_integer_of_nat
tff(fact_5522_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_integer(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),L) ).
% fst_divmod_integer
tff(fact_5523_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(ord_min_Code_integer(K),L)) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ).
% int_of_integer_min
tff(fact_5524_int__of__integer__integer__of__nat,axiom,
! [N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% int_of_integer_integer_of_nat
tff(fact_5525_integer__of__nat_Orep__eq,axiom,
! [X: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,X)) = aa_nat_int(semiri1314217659103216013at_int,X) ).
% integer_of_nat.rep_eq
tff(fact_5526_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_abs(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ).
% fst_divmod_abs
tff(fact_5527_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)) = product_fst_nat_nat(unique5055182867167087721od_nat(one,K)) ).
% Suc_0_div_numeral
tff(fact_5528_inf__int__def,axiom,
inf_inf_int = ord_min_int ).
% inf_int_def
tff(fact_5529_integer__of__nat__eq__of__nat,axiom,
code_integer_of_nat = semiri4939895301339042750nteger ).
% integer_of_nat_eq_of_nat
tff(fact_5530_integer__of__nat__0,axiom,
aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).
% integer_of_nat_0
tff(fact_5531_integer__of__nat_Oabs__eq,axiom,
! [X: nat] : aa_nat_Code_integer(code_integer_of_nat,X) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,X)) ).
% integer_of_nat.abs_eq
tff(fact_5532_integer__of__nat__numeral,axiom,
! [N: num] : aa_nat_Code_integer(code_integer_of_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_Code_integer(numera6620942414471956472nteger,N) ).
% integer_of_nat_numeral
tff(fact_5533_integer__of__nat__1,axiom,
aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).
% integer_of_nat_1
tff(fact_5534_rat__sgn__code,axiom,
! [P3: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P3)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P3)))),one_one_int) ).
% rat_sgn_code
tff(fact_5535_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( bezw(X,Xa2) = Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).
% bezw.elims
tff(fact_5536_bezw_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero_nat )
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
& ( ( Y != zero_zero_nat )
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).
% bezw.simps
tff(fact_5537_quotient__of__denom__pos_H,axiom,
! [R: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R)))) ).
% quotient_of_denom_pos'
tff(fact_5538_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).
% bezw_non_0
tff(fact_5539_minus__one__mod__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = adjust_mod(aa_num_int(numeral_numeral_int,N),product_snd_int_int(unique5052692396658037445od_int(one,N))) ).
% minus_one_mod_numeral
tff(fact_5540_one__mod__minus__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_mod(aa_num_int(numeral_numeral_int,N),product_snd_int_int(unique5052692396658037445od_int(one,N)))) ).
% one_mod_minus_numeral
tff(fact_5541_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,K),L) ).
% snd_divmod_integer
tff(fact_5542_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_abs(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ).
% snd_divmod_abs
tff(fact_5543_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)) = product_snd_nat_nat(unique5055182867167087721od_nat(one,K)) ).
% Suc_0_mod_numeral
tff(fact_5544_Divides_Oadjust__mod__def,axiom,
! [R: int,L: int] :
( ( ( R = zero_zero_int )
=> ( adjust_mod(L,R) = zero_zero_int ) )
& ( ( R != zero_zero_int )
=> ( adjust_mod(L,R) = aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R) ) ) ) ).
% Divides.adjust_mod_def
tff(fact_5545_positive__def,axiom,
positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_fk) ).
% positive_def
tff(fact_5546_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( bezw(X,Xa2) = Y )
=> ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
=> ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).
% bezw.pelims
tff(fact_5547_positive_Orep__eq,axiom,
! [X: rat] :
( pp(aa_rat_bool(positive,X))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).
% positive.rep_eq
tff(fact_5548_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
=> ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
=> ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
=> ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_5549_plus__rat__def,axiom,
plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_fl) ).
% plus_rat_def
tff(fact_5550_inverse__rat__def,axiom,
inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_fm) ).
% inverse_rat_def
tff(fact_5551_one__rat__def,axiom,
one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) ).
% one_rat_def
tff(fact_5552_Fract_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_int_rat(aa_int_fun_int_rat(fract,Xa2),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Xa2),X))) ).
% Fract.abs_eq
tff(fact_5553_zero__rat__def,axiom,
zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)) ).
% zero_rat_def
tff(fact_5554_Fract__def,axiom,
fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_fn) ).
% Fract_def
tff(fact_5555_uminus__rat__def,axiom,
uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_fo) ).
% uminus_rat_def
tff(fact_5556_times__rat__def,axiom,
times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_fp) ).
% times_rat_def
tff(fact_5557_normalize__def,axiom,
! [P3: product_prod_int_int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
=> ( normalize(P3) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
=> ( ( ( product_snd_int_int(P3) = zero_zero_int )
=> ( normalize(P3) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ) )
& ( ( product_snd_int_int(P3) != zero_zero_int )
=> ( normalize(P3) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))) ) ) ) ) ) ).
% normalize_def
tff(fact_5558_of__int__eq__id,axiom,
ring_1_of_int_int = id_int ).
% of_int_eq_id
tff(fact_5559_gcd__1__int,axiom,
! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).
% gcd_1_int
tff(fact_5560_gcd__neg1__int,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).
% gcd_neg1_int
tff(fact_5561_gcd__neg2__int,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).
% gcd_neg2_int
tff(fact_5562_gcd__abs2__int,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(abs_abs_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).
% gcd_abs2_int
tff(fact_5563_gcd__abs1__int,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(abs_abs_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).
% gcd_abs1_int
tff(fact_5564_abs__gcd__int,axiom,
! [X: int,Y: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).
% abs_gcd_int
tff(fact_5565_gcd__pos__int,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)))
<=> ( ( M != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
% gcd_pos_int
tff(fact_5566_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,N)) ).
% gcd_neg_numeral_2_int
tff(fact_5567_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),X) ).
% gcd_neg_numeral_1_int
tff(fact_5568_gcd__0__int,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = aa_int_int(abs_abs_int,X) ).
% gcd_0_int
tff(fact_5569_gcd__0__left__int,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = aa_int_int(abs_abs_int,X) ).
% gcd_0_left_int
tff(fact_5570_gcd__proj2__if__dvd__int,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
=> ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(abs_abs_int,Y) ) ) ).
% gcd_proj2_if_dvd_int
tff(fact_5571_gcd__proj1__if__dvd__int,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
=> ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(abs_abs_int,X) ) ) ).
% gcd_proj1_if_dvd_int
tff(fact_5572_gcd__idem__int,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),X) = aa_int_int(abs_abs_int,X) ).
% gcd_idem_int
tff(fact_5573_gcd__red__int,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ).
% gcd_red_int
tff(fact_5574_gcd__ge__0__int,axiom,
! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ).
% gcd_ge_0_int
tff(fact_5575_bezout__int,axiom,
! [X: int,Y: int] :
? [U3: int,V3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V3),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).
% bezout_int
tff(fact_5576_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N)) ).
% gcd_mult_distrib_int
tff(fact_5577_gcd__le2__int,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B)) ) ).
% gcd_le2_int
tff(fact_5578_gcd__le1__int,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A)) ) ).
% gcd_le1_int
tff(fact_5579_gcd__cases__int,axiom,
! [X: int,Y: int,P: fun_int_bool] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ) ) ).
% gcd_cases_int
tff(fact_5580_gcd__unique__int,axiom,
! [D: int,A: int,B: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),A))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
& ! [E4: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E4),A))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E4),B)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E4),D)) ) )
<=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).
% gcd_unique_int
tff(fact_5581_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
=> ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ) ) ).
% gcd_non_0_int
tff(fact_5582_gcd__code__int,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,L),zero_zero_int),K,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).
% gcd_code_int
tff(fact_5583_Code__Numeral_Odup__def,axiom,
code_dup = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),aTP_Lamm_fq) ).
% Code_Numeral.dup_def
tff(fact_5584_gcd__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.eq_neutr_iff
tff(fact_5585_gcd__nat_Oleft__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).
% gcd_nat.left_neutral
tff(fact_5586_gcd__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
<=> ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.neutr_eq_iff
tff(fact_5587_gcd__nat_Oright__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).
% gcd_nat.right_neutral
tff(fact_5588_gcd__0__nat,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).
% gcd_0_nat
tff(fact_5589_gcd__0__left__nat,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).
% gcd_0_left_nat
tff(fact_5590_gcd__1__nat,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).
% gcd_1_nat
tff(fact_5591_gcd__nat_Oabsorb1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).
% gcd_nat.absorb1
tff(fact_5592_gcd__nat_Oabsorb2,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).
% gcd_nat.absorb2
tff(fact_5593_gcd__nat_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% gcd_nat.bounded_iff
tff(fact_5594_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ) ).
% gcd_proj1_if_dvd_nat
tff(fact_5595_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y),X))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ) ).
% gcd_proj2_if_dvd_nat
tff(fact_5596_gcd__Suc__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).
% gcd_Suc_0
tff(fact_5597_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
<=> ( ( M != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
tff(fact_5598_gcd__int__int__eq,axiom,
! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) ).
% gcd_int_int_eq
tff(fact_5599_gcd__nat__abs__left__eq,axiom,
! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% gcd_nat_abs_left_eq
tff(fact_5600_gcd__nat__abs__right__eq,axiom,
! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ).
% gcd_nat_abs_right_eq
tff(fact_5601_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).
% gcd_mult_distrib_nat
tff(fact_5602_gcd__nat_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D))) ) ) ).
% gcd_nat.mono
tff(fact_5603_gcd__nat_OorderE,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).
% gcd_nat.orderE
tff(fact_5604_gcd__nat_OorderI,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% gcd_nat.orderI
tff(fact_5605_gcd__nat_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).
% gcd_nat.absorb3
tff(fact_5606_gcd__nat_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
& ( B != A ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).
% gcd_nat.absorb4
tff(fact_5607_gcd__nat_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% gcd_nat.boundedE
tff(fact_5608_gcd__nat_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))) ) ) ).
% gcd_nat.boundedI
tff(fact_5609_gcd__nat_Oorder__iff,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).
% gcd_nat.order_iff
tff(fact_5610_gcd__nat_Ocobounded1,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ).
% gcd_nat.cobounded1
tff(fact_5611_gcd__nat_Ocobounded2,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ).
% gcd_nat.cobounded2
tff(fact_5612_gcd__nat_Oabsorb__iff1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).
% gcd_nat.absorb_iff1
tff(fact_5613_gcd__nat_Oabsorb__iff2,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).
% gcd_nat.absorb_iff2
tff(fact_5614_gcd__nat_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)) ) ).
% gcd_nat.coboundedI1
tff(fact_5615_gcd__nat_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)) ) ).
% gcd_nat.coboundedI2
tff(fact_5616_gcd__nat_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
& ( A != aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) )
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
& ( A != C ) ) ) ) ).
% gcd_nat.strict_boundedE
tff(fact_5617_gcd__nat_Ostrict__order__iff,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ( A != B ) )
<=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
& ( A != B ) ) ) ).
% gcd_nat.strict_order_iff
tff(fact_5618_gcd__nat_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
& ( A != C ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C))
& ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).
% gcd_nat.strict_coboundedI1
tff(fact_5619_gcd__nat_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
& ( B != C ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C))
& ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).
% gcd_nat.strict_coboundedI2
tff(fact_5620_gcd__unique__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
& ! [E4: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E4),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E4),B)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E4),D)) ) )
<=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).
% gcd_unique_nat
tff(fact_5621_gcd__red__nat,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ).
% gcd_red_nat
tff(fact_5622_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).
% gcd_le1_nat
tff(fact_5623_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ) ).
% gcd_le2_nat
tff(fact_5624_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).
% gcd_diff2_nat
tff(fact_5625_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).
% gcd_diff1_nat
tff(fact_5626_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).
% gcd_non_0_nat
tff(fact_5627_gcd__nat_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
& ( ( Y != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).
% gcd_nat.simps
tff(fact_5628_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) ) ) ).
% gcd_nat.elims
tff(fact_5629_gcd__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,X),Xa2)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).
% gcd_integer.rep_eq
tff(fact_5630_gcd__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa2),X)) ).
% gcd_integer.abs_eq
tff(fact_5631_Code__Numeral_Odup__code_I1_J,axiom,
aa_Cod1783974652533396728nteger(code_dup,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% Code_Numeral.dup_code(1)
tff(fact_5632_Gcd__in,axiom,
! [A3: set_nat] :
( ! [A2: nat,B2: nat] :
( pp(aa_set_nat_bool(member_nat(A2),A3))
=> ( pp(aa_set_nat_bool(member_nat(B2),A3))
=> pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2)),A3)) ) )
=> ( ( A3 != bot_bot_set_nat )
=> pp(aa_set_nat_bool(member_nat(gcd_Gcd_nat(A3)),A3)) ) ) ).
% Gcd_in
tff(fact_5633_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X2: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).
% bezout_nat
tff(fact_5634_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X2: nat,Y2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)))
& ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)))
& ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).
% bezout_gcd_nat'
tff(fact_5635_gcd__code__integer,axiom,
! [K: code_integer,L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,K),L) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,L),zero_z3403309356797280102nteger),K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))))) ).
% gcd_code_integer
tff(fact_5636_dup_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(code_dup,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,X)) ).
% dup.rep_eq
tff(fact_5637_dup_Oabs__eq,axiom,
! [X: int] : aa_Cod1783974652533396728nteger(code_dup,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)) ).
% dup.abs_eq
tff(fact_5638_gcd__int__def,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).
% gcd_int_def
tff(fact_5639_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_cd) ).
% gcd_nat.semilattice_neutr_order_axioms
tff(fact_5640_gcd__integer__def,axiom,
gcd_gcd_Code_integer = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),gcd_gcd_int) ).
% gcd_integer_def
tff(fact_5641_bezw__aux,axiom,
! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),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(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).
% bezw_aux
tff(fact_5642_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
=> ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) )
=> ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).
% gcd_nat.pelims
tff(fact_5643_plus__rat_Oabs__eq,axiom,
! [Xa2: product_prod_int_int,X: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa2),Xa2))
=> ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_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,product_fst_int_int(Xa2)),product_snd_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X)))) ) ) ) ).
% plus_rat.abs_eq
tff(fact_5644_gcd__idem__nat,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).
% gcd_idem_nat
tff(fact_5645_gcd__nat_Oright__idem,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).
% gcd_nat.right_idem
tff(fact_5646_gcd__nat_Oleft__idem,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).
% gcd_nat.left_idem
tff(fact_5647_gcd__nat_Oidem,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),A) = A ).
% gcd_nat.idem
tff(fact_5648_ratrel__iff,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),Y))
<=> ( ( product_snd_int_int(X) != zero_zero_int )
& ( product_snd_int_int(Y) != zero_zero_int )
& ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Y)),product_snd_int_int(X)) ) ) ) ).
% ratrel_iff
tff(fact_5649_one__rat_Orsp,axiom,
pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int))) ).
% one_rat.rsp
tff(fact_5650_zero__rat_Orsp,axiom,
pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int))) ).
% zero_rat.rsp
tff(fact_5651_ratrel__def,axiom,
! [X3: product_prod_int_int,Xa: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X3),Xa))
<=> ( ( product_snd_int_int(X3) != zero_zero_int )
& ( product_snd_int_int(Xa) != zero_zero_int )
& ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X3)),product_snd_int_int(Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa)),product_snd_int_int(X3)) ) ) ) ).
% ratrel_def
tff(fact_5652_positive_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
=> ( pp(aa_rat_bool(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(X)))) ) ) ).
% positive.abs_eq
tff(fact_5653_inverse__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
=> ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(X)),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(X)),product_fst_int_int(X)))) ) ) ).
% inverse_rat.abs_eq
tff(fact_5654_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N))),one_one_Code_integer) ).
% Code_Numeral.sub_code(9)
tff(fact_5655_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N))),one_one_Code_integer) ).
% Code_Numeral.sub_code(8)
tff(fact_5656_Code__Numeral_Osub__code_I1_J,axiom,
aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),one) = zero_z3403309356797280102nteger ).
% Code_Numeral.sub_code(1)
tff(fact_5657_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N)) ).
% Code_Numeral.sub_code(6)
tff(fact_5658_Code__Numeral_Osub__code_I7_J,axiom,
! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N)) ).
% Code_Numeral.sub_code(7)
tff(fact_5659_sub_Orep__eq,axiom,
! [X: num,Xa2: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,X),Xa2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Xa2)) ).
% sub.rep_eq
tff(fact_5660_sub_Oabs__eq,axiom,
! [Xa2: num,X: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,Xa2),X) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Xa2)),aa_num_int(numeral_numeral_int,X))) ).
% sub.abs_eq
tff(fact_5661_Code__Numeral_Osub__def,axiom,
code_sub = map_fu8540202848259802501nteger(id_num,map_fu8638147718074629079nteger(id_num,code_integer_of_int),aTP_Lamm_fr) ).
% Code_Numeral.sub_def
tff(fact_5662_cr__rat__def,axiom,
! [X3: product_prod_int_int,Xa: rat] :
( pp(aa_rat_bool(cr_rat(X3),Xa))
<=> ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X3),X3))
& ( aa_Pro6166618844887768661nt_rat(abs_Rat,X3) = Xa ) ) ) ).
% cr_rat_def
tff(fact_5663_card__UNIV__unit,axiom,
finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).
% card_UNIV_unit
tff(fact_5664_infinite__UNIV__int,axiom,
~ finite_finite_int(top_top_set_int) ).
% infinite_UNIV_int
tff(fact_5665_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y2: int] :
( ( X = aa_int_Code_integer(code_integer_of_int,Y2) )
=> ~ member_int(Y2,top_top_set_int) ) ).
% integer_of_int_cases
tff(fact_5666_integer__of__int__induct,axiom,
! [P: fun_Co8822979462182099027r_bool,X: code_integer] :
( ! [Y2: int] :
( member_int(Y2,top_top_set_int)
=> pp(aa_Code_integer_bool(P,aa_int_Code_integer(code_integer_of_int,Y2))) )
=> pp(aa_Code_integer_bool(P,X)) ) ).
% integer_of_int_induct
tff(fact_5667_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( member_int(X,top_top_set_int)
=> ( member_int(Y,top_top_set_int)
=> ( ( aa_int_Code_integer(code_integer_of_int,X) = aa_int_Code_integer(code_integer_of_int,Y) )
<=> ( X = Y ) ) ) ) ).
% integer_of_int_inject
tff(fact_5668_int__of__integer,axiom,
! [X: code_integer] : member_int(aa_Code_integer_int(code_int_of_integer,X),top_top_set_int) ).
% int_of_integer
tff(fact_5669_int__of__integer__cases,axiom,
! [Y: int] :
( member_int(Y,top_top_set_int)
=> ~ ! [X2: code_integer] : Y != aa_Code_integer_int(code_int_of_integer,X2) ) ).
% int_of_integer_cases
tff(fact_5670_int__of__integer__induct,axiom,
! [Y: int,P: fun_int_bool] :
( member_int(Y,top_top_set_int)
=> ( ! [X2: code_integer] : pp(aa_int_bool(P,aa_Code_integer_int(code_int_of_integer,X2)))
=> pp(aa_int_bool(P,Y)) ) ) ).
% int_of_integer_induct
tff(fact_5671_integer__of__int__inverse,axiom,
! [Y: int] :
( member_int(Y,top_top_set_int)
=> ( aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,Y)) = Y ) ) ).
% integer_of_int_inverse
tff(fact_5672_int__in__range__abs,axiom,
! [N: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,N),image_int_int(abs_abs_int,top_top_set_int)) ).
% int_in_range_abs
tff(fact_5673_range__mod,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fs,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).
% range_mod
tff(fact_5674_UN__atMost__UNIV,axiom,
comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).
% UN_atMost_UNIV
tff(fact_5675_UN__lessThan__UNIV,axiom,
comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).
% UN_lessThan_UNIV
tff(fact_5676_Sup__nat__empty,axiom,
complete_Sup_Sup_nat(bot_bot_set_nat) = zero_zero_nat ).
% Sup_nat_empty
tff(fact_5677_natLess__def,axiom,
bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).
% natLess_def
tff(fact_5678_Gcd__eq__Max,axiom,
! [M2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,M2))
=> ( ( M2 != bot_bot_set_nat )
=> ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M2))
=> ( gcd_Gcd_nat(M2) = lattic8265883725875713057ax_nat(comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_ft,M2))) ) ) ) ) ).
% Gcd_eq_Max
tff(fact_5679_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cs,N))) = N ) ) ).
% Max_divisors_self_nat
tff(fact_5680_Sup__nat__def,axiom,
! [X4: set_nat] :
( ( ( X4 = bot_bot_set_nat )
=> ( complete_Sup_Sup_nat(X4) = zero_zero_nat ) )
& ( ( X4 != bot_bot_set_nat )
=> ( complete_Sup_Sup_nat(X4) = lattic8265883725875713057ax_nat(X4) ) ) ) ).
% Sup_nat_def
tff(fact_5681_card__le__Suc__Max,axiom,
! [S3: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(S3)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S3)))) ) ).
% card_le_Suc_Max
tff(fact_5682_divide__nat__def,axiom,
! [N: nat,M: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fu(N),M))) ) ) ) ).
% divide_nat_def
tff(fact_5683_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fv(N),M))) ) ) ).
% gcd_is_Max_divisors_nat
tff(fact_5684_Max__divisors__self__int,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_cx,N))) = aa_int_int(abs_abs_int,N) ) ) ).
% Max_divisors_self_int
tff(fact_5685_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_fw(N),M))) ) ) ).
% gcd_is_Max_divisors_int
tff(fact_5686_INT__greaterThan__UNIV,axiom,
comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).
% INT_greaterThan_UNIV
tff(fact_5687_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).
% atLeastLessThan_add_Un
tff(fact_5688_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_5689_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_5690_atLeast__0,axiom,
aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).
% atLeast_0
tff(fact_5691_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
tff(fact_5692_sup__int__def,axiom,
sup_sup_int = ord_max_int ).
% sup_int_def
tff(fact_5693_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_5694_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_5695_UN__atLeast__UNIV,axiom,
comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).
% UN_atLeast_UNIV
tff(fact_5696_range__abs__Nats,axiom,
image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).
% range_abs_Nats
tff(fact_5697_binomial__def,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_fx(N),K))) ).
% binomial_def
tff(fact_5698_pred__nat__def,axiom,
pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_fy)) ).
% pred_nat_def
tff(fact_5699_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi2905341329935302413cl_nat(pred_nat))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% pred_nat_trancl_eq_le
tff(fact_5700_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_5701_mono__Suc,axiom,
order_mono_nat_nat(suc) ).
% mono_Suc
tff(fact_5702_mono__times__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).
% mono_times_nat
tff(fact_5703_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fz,K)) ) ).
% mono_ge2_power_minus_self
tff(fact_5704_quotient__of__def,axiom,
! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_ga(X)) ).
% quotient_of_def
tff(fact_5705_normalize__stable,axiom,
! [Q2: int,P3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P3),Q2))
=> ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q2) ) ) ) ).
% normalize_stable
tff(fact_5706_inj__on__diff__nat,axiom,
! [N4: set_nat,K: nat] :
( ! [N2: nat] :
( pp(aa_set_nat_bool(member_nat(N2),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
=> pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,K)),N4)) ) ).
% inj_on_diff_nat
tff(fact_5707_coprime__crossproduct__int,axiom,
! [A: int,D: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(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_5708_inj__Suc,axiom,
! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).
% inj_Suc
tff(fact_5709_Rat__cases,axiom,
! [Q2: rat] :
~ ! [A2: int,B2: int] :
( ( Q2 = aa_int_rat(aa_int_fun_int_rat(fract,A2),B2) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A2),B2)) ) ) ).
% Rat_cases
tff(fact_5710_Rat__induct,axiom,
! [P: fun_rat_bool,Q2: rat] :
( ! [A2: int,B2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A2),B2))
=> pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A2),B2))) ) )
=> pp(aa_rat_bool(P,Q2)) ) ).
% Rat_induct
tff(fact_5711_coprime__common__divisor__int,axiom,
! [A: int,B: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),B))
=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).
% coprime_common_divisor_int
tff(fact_5712_Rat__cases__nonzero,axiom,
! [Q2: rat] :
( ! [A2: int,B2: int] :
( ( Q2 = aa_int_rat(aa_int_fun_int_rat(fract,A2),B2) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
=> ( ( A2 != zero_zero_int )
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A2),B2)) ) ) )
=> ( Q2 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
tff(fact_5713_quotient__of__unique,axiom,
! [R: rat] :
? [X2: product_prod_int_int] :
( ( R = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(X2)),product_snd_int_int(X2)) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(X2)))
& pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(X2)),product_snd_int_int(X2)))
& ! [Y3: product_prod_int_int] :
( ( ( R = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Y3)),product_snd_int_int(Y3)) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Y3)))
& pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Y3)),product_snd_int_int(Y3))) )
=> ( Y3 = X2 ) ) ) ).
% quotient_of_unique
tff(fact_5714_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M),N)) ) ).
% coprime_int_iff
tff(fact_5715_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
<=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% coprime_nat_abs_left_iff
tff(fact_5716_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).
% coprime_nat_abs_right_iff
tff(fact_5717_coprime__common__divisor__nat,axiom,
! [A: nat,B: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),B))
=> ( X = one_one_nat ) ) ) ) ).
% coprime_common_divisor_nat
tff(fact_5718_coprime__Suc__right__nat,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,N))) ).
% coprime_Suc_right_nat
tff(fact_5719_coprime__Suc__left__nat,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,N)),N)) ).
% coprime_Suc_left_nat
tff(fact_5720_coprime__crossproduct__nat,axiom,
! [A: nat,D: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,B),C))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D) )
<=> ( ( A = B )
& ( C = D ) ) ) ) ) ).
% coprime_crossproduct_nat
tff(fact_5721_coprime__Suc__0__left,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).
% coprime_Suc_0_left
tff(fact_5722_coprime__Suc__0__right,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).
% coprime_Suc_0_right
tff(fact_5723_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).
% coprime_diff_one_left_nat
tff(fact_5724_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).
% coprime_diff_one_right_nat
tff(fact_5725_positive_Orsp,axiom,
bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_fk,aTP_Lamm_fk) ).
% positive.rsp
tff(fact_5726_Fract_Orsp,axiom,
bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_fn,aTP_Lamm_fn) ).
% Fract.rsp
tff(fact_5727_less__eq__integer_Orsp,axiom,
bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_eq_int,ord_less_eq_int) ).
% less_eq_integer.rsp
tff(fact_5728_less__eq__natural_Orsp,axiom,
bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_eq_nat,ord_less_eq_nat) ).
% less_eq_natural.rsp
tff(fact_5729_uminus__integer_Orsp,axiom,
pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),uminus_uminus_int),uminus_uminus_int)) ).
% uminus_integer.rsp
tff(fact_5730_minus__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),minus_minus_nat,minus_minus_nat) ).
% minus_natural.rsp
tff(fact_5731_minus__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),minus_minus_int,minus_minus_int) ).
% minus_integer.rsp
tff(fact_5732_sub_Orsp,axiom,
bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_fr,aTP_Lamm_fr) ).
% sub.rsp
tff(fact_5733_gcd__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),gcd_gcd_int,gcd_gcd_int) ).
% gcd_integer.rsp
tff(fact_5734_integer__of__natural_Orsp,axiom,
bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,semiri1314217659103216013at_int,semiri1314217659103216013at_int) ).
% integer_of_natural.rsp
tff(fact_5735_dup_Orsp,axiom,
pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_fq),aTP_Lamm_fq)) ).
% dup.rsp
tff(fact_5736_sgn__integer_Orsp,axiom,
pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),sgn_sgn_int),sgn_sgn_int)) ).
% sgn_integer.rsp
tff(fact_5737_plus__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).
% plus_natural.rsp
tff(fact_5738_plus__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).
% plus_integer.rsp
tff(fact_5739_bit__natural_Orsp,axiom,
bNF_re1480548845665215743t_bool(fequal_nat,fequal_fun_nat_bool,bit_se1148574629649215175it_nat,bit_se1148574629649215175it_nat) ).
% bit_natural.rsp
tff(fact_5740_Suc_Orsp,axiom,
pp(aa_fun_nat_nat_bool(aa_fun3185460345579526648t_bool(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc)) ).
% Suc.rsp
tff(fact_5741_natural__of__integer_Orsp,axiom,
bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,nat2,nat2) ).
% natural_of_integer.rsp
tff(fact_5742_not__integer_Orsp,axiom,
pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_ri7919022796975470100ot_int),bit_ri7919022796975470100ot_int)) ).
% not_integer.rsp
tff(fact_5743_and__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se727722235901077358nd_nat,bit_se727722235901077358nd_nat) ).
% and_natural.rsp
tff(fact_5744_and__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se725231765392027082nd_int,bit_se725231765392027082nd_int) ).
% and_integer.rsp
tff(fact_5745_take__bit__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2925701944663578781it_nat,bit_se2925701944663578781it_nat) ).
% take_bit_natural.rsp
tff(fact_5746_take__bit__integer_Orsp,axiom,
bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se2923211474154528505it_int,bit_se2923211474154528505it_int) ).
% take_bit_integer.rsp
tff(fact_5747_modulo__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),modulo_modulo_nat,modulo_modulo_nat) ).
% modulo_natural.rsp
tff(fact_5748_modulo__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),modulo_modulo_int,modulo_modulo_int) ).
% modulo_integer.rsp
tff(fact_5749_unset__bit__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se4205575877204974255it_nat,bit_se4205575877204974255it_nat) ).
% unset_bit_natural.rsp
tff(fact_5750_unset__bit__integer_Orsp,axiom,
bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se4203085406695923979it_int,bit_se4203085406695923979it_int) ).
% unset_bit_integer.rsp
tff(fact_5751_set__bit__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se7882103937844011126it_nat,bit_se7882103937844011126it_nat) ).
% set_bit_natural.rsp
tff(fact_5752_set__bit__integer_Orsp,axiom,
bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se7879613467334960850it_int,bit_se7879613467334960850it_int) ).
% set_bit_integer.rsp
tff(fact_5753_flip__bit__integer_Orsp,axiom,
bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se2159334234014336723it_int,bit_se2159334234014336723it_int) ).
% flip_bit_integer.rsp
tff(fact_5754_flip__bit__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2161824704523386999it_nat,bit_se2161824704523386999it_nat) ).
% flip_bit_natural.rsp
tff(fact_5755_push__bit__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se547839408752420682it_nat,bit_se547839408752420682it_nat) ).
% push_bit_natural.rsp
tff(fact_5756_push__bit__integer_Orsp,axiom,
bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se545348938243370406it_int,bit_se545348938243370406it_int) ).
% push_bit_integer.rsp
tff(fact_5757_xor__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se6528837805403552850or_nat,bit_se6528837805403552850or_nat) ).
% xor_natural.rsp
tff(fact_5758_xor__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se6526347334894502574or_int,bit_se6526347334894502574or_int) ).
% xor_integer.rsp
tff(fact_5759_bit__integer_Orsp,axiom,
bNF_re4512597724629091767t_bool(fequal_int,fequal_fun_nat_bool,bit_se1146084159140164899it_int,bit_se1146084159140164899it_int) ).
% bit_integer.rsp
tff(fact_5760_or__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se1409905431419307370or_int,bit_se1409905431419307370or_int) ).
% or_integer.rsp
tff(fact_5761_or__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se1412395901928357646or_nat,bit_se1412395901928357646or_nat) ).
% or_natural.rsp
tff(fact_5762_drop__bit__integer_Orsp,axiom,
bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se8568078237143864401it_int,bit_se8568078237143864401it_int) ).
% drop_bit_integer.rsp
tff(fact_5763_drop__bit__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se8570568707652914677it_nat,bit_se8570568707652914677it_nat) ).
% drop_bit_natural.rsp
tff(fact_5764_mask__integer_Orsp,axiom,
bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,bit_se2000444600071755411sk_int,bit_se2000444600071755411sk_int) ).
% mask_integer.rsp
tff(fact_5765_mask__natural_Orsp,axiom,
pp(aa_fun_nat_nat_bool(aa_fun3185460345579526648t_bool(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2002935070580805687sk_nat),bit_se2002935070580805687sk_nat)) ).
% mask_natural.rsp
tff(fact_5766_abs__integer_Orsp,axiom,
pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),abs_abs_int),abs_abs_int)) ).
% abs_integer.rsp
tff(fact_5767_divide__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),divide_divide_int,divide_divide_int) ).
% divide_integer.rsp
tff(fact_5768_divide__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),divide_divide_nat,divide_divide_nat) ).
% divide_natural.rsp
tff(fact_5769_times__natural_Orsp,axiom,
bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).
% times_natural.rsp
tff(fact_5770_times__integer_Orsp,axiom,
bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).
% times_integer.rsp
tff(fact_5771_less__integer_Orsp,axiom,
bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_int,ord_less_int) ).
% less_integer.rsp
tff(fact_5772_less__natural_Orsp,axiom,
bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).
% less_natural.rsp
tff(fact_5773_times__rat_Orsp,axiom,
bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_fp,aTP_Lamm_fp) ).
% times_rat.rsp
tff(fact_5774_plus__rat_Orsp,axiom,
bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_fl,aTP_Lamm_fl) ).
% plus_rat.rsp
tff(fact_5775_num__of__integer_Orsp,axiom,
bNF_re7626690874201225453um_num(fequal_int,fequal_num,comp_nat_num_int(num_of_nat,nat2),comp_nat_num_int(num_of_nat,nat2)) ).
% num_of_integer.rsp
tff(fact_5776_uminus__rat_Orsp,axiom,
pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_fo),aTP_Lamm_fo)) ).
% uminus_rat.rsp
tff(fact_5777_inverse__rat_Orsp,axiom,
pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_fm),aTP_Lamm_fm)) ).
% inverse_rat.rsp
tff(fact_5778_plus__rat_Otransfer,axiom,
bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_fl,plus_plus_rat) ).
% plus_rat.transfer
tff(fact_5779_inverse__rat_Otransfer,axiom,
pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_fm),inverse_inverse_rat)) ).
% inverse_rat.transfer
tff(fact_5780_one__rat_Otransfer,axiom,
pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),one_one_rat)) ).
% one_rat.transfer
tff(fact_5781_zero__rat_Otransfer,axiom,
pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),zero_zero_rat)) ).
% zero_rat.transfer
tff(fact_5782_Fract_Otransfer,axiom,
bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_fn,fract) ).
% Fract.transfer
tff(fact_5783_uminus__rat_Otransfer,axiom,
pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_fo),uminus_uminus_rat)) ).
% uminus_rat.transfer
tff(fact_5784_times__rat_Otransfer,axiom,
bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_fp,times_times_rat) ).
% times_rat.transfer
tff(fact_5785_positive_Otransfer,axiom,
bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_fk,positive) ).
% positive.transfer
tff(fact_5786_times__int_Otransfer,axiom,
bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ec),times_times_int) ).
% times_int.transfer
tff(fact_5787_zero__int_Otransfer,axiom,
pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).
% zero_int.transfer
tff(fact_5788_int__transfer,axiom,
bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_gb,semiri1314217659103216013at_int) ).
% int_transfer
tff(fact_5789_uminus__int_Otransfer,axiom,
pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_ed)),uminus_uminus_int)) ).
% uminus_int.transfer
tff(fact_5790_nat_Otransfer,axiom,
bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).
% nat.transfer
tff(fact_5791_one__int_Otransfer,axiom,
pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).
% one_int.transfer
tff(fact_5792_less__int_Otransfer,axiom,
bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_ef),ord_less_int) ).
% less_int.transfer
tff(fact_5793_less__eq__int_Otransfer,axiom,
bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_eh),ord_less_eq_int) ).
% less_eq_int.transfer
tff(fact_5794_plus__int_Otransfer,axiom,
bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ej),plus_plus_int) ).
% plus_int.transfer
tff(fact_5795_minus__int_Otransfer,axiom,
bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_el),minus_minus_int) ).
% minus_int.transfer
tff(fact_5796_times__int_Orsp,axiom,
bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ec),produc4528391901023682385at_nat(aTP_Lamm_ec)) ).
% times_int.rsp
tff(fact_5797_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V)))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).
% intrel_iff
tff(fact_5798_int_Orel__eq__transfer,axiom,
bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),intrel,fequal_int) ).
% int.rel_eq_transfer
tff(fact_5799_zero__int_Orsp,axiom,
pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).
% zero_int.rsp
tff(fact_5800_int_Oabs__eq__iff,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( aa_Pro4240683299535347697at_int(abs_Integ,X) = aa_Pro4240683299535347697at_int(abs_Integ,Y) )
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,X),Y)) ) ).
% int.abs_eq_iff
tff(fact_5801_uminus__int_Orsp,axiom,
pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_ed)),produc2626176000494625587at_nat(aTP_Lamm_ed))) ).
% uminus_int.rsp
tff(fact_5802_nat_Orsp,axiom,
bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).
% nat.rsp
tff(fact_5803_one__int_Orsp,axiom,
pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).
% one_int.rsp
tff(fact_5804_intrel__def,axiom,
intrel = produc5023667602222594023t_bool(aTP_Lamm_gd) ).
% intrel_def
tff(fact_5805_less__int_Orsp,axiom,
bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_ef),produc5023667602222594023t_bool(aTP_Lamm_ef)) ).
% less_int.rsp
tff(fact_5806_less__eq__int_Orsp,axiom,
bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_eh),produc5023667602222594023t_bool(aTP_Lamm_eh)) ).
% less_eq_int.rsp
tff(fact_5807_minus__int_Orsp,axiom,
bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_el),produc4528391901023682385at_nat(aTP_Lamm_el)) ).
% minus_int.rsp
tff(fact_5808_plus__int_Orsp,axiom,
bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ej),produc4528391901023682385at_nat(aTP_Lamm_ej)) ).
% plus_int.rsp
tff(fact_5809_le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S3),N))) ) ).
% le_enumerate
tff(fact_5810_finite__le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite_card_nat(S3)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S3),N))) ) ) ).
% finite_le_enumerate
tff(fact_5811_Least__eq__0,axiom,
! [P: fun_nat_bool] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ord_Least_nat(P) = zero_zero_nat ) ) ).
% Least_eq_0
tff(fact_5812_Least__Suc2,axiom,
! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
( pp(aa_nat_bool(P,N))
=> ( pp(aa_nat_bool(Q,M))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [K3: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(suc,K3)))
<=> pp(aa_nat_bool(Q,K3)) )
=> ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).
% Least_Suc2
tff(fact_5813_Least__Suc,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,N))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_ge(P))) ) ) ) ).
% Least_Suc
tff(fact_5814_Inf__nat__def,axiom,
! [X4: set_nat] : complete_Inf_Inf_nat(X4) = ord_Least_nat(aTP_Lamm_ak(X4)) ).
% Inf_nat_def
tff(fact_5815_type__definition__integer,axiom,
type_d8366093980585677751er_int(code_int_of_integer,code_integer_of_int,top_top_set_int) ).
% type_definition_integer
tff(fact_5816_mset__set__upto__eq__mset__upto,axiom,
! [N: nat] : mset_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N)) = mset_nat(upt(zero_zero_nat,N)) ).
% mset_set_upto_eq_mset_upto
tff(fact_5817_filterlim__lessThan__at__top,axiom,
filter3212408913953519116et_nat(set_ord_lessThan_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).
% filterlim_lessThan_at_top
tff(fact_5818_filterlim__atMost__at__top,axiom,
filter3212408913953519116et_nat(set_ord_atMost_nat,finite3254316476582989077op_nat(top_top_set_nat),at_top_nat) ).
% filterlim_atMost_at_top
tff(fact_5819_bij__betw__Suc,axiom,
! [M2: set_nat,N4: set_nat] :
( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M2),N4))
<=> ( image_nat_nat(suc,M2) = N4 ) ) ).
% bij_betw_Suc
tff(fact_5820_Sup__int__def,axiom,
! [X4: set_int] : complete_Sup_Sup_int(X4) = the_int(aTP_Lamm_gf(X4)) ).
% Sup_int_def
tff(fact_5821_and__not__num_Opelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( bit_and_not_num(X,Xa2) = Y )
=> ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(X,Xa2))
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_dd,bit_and_not_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
tff(fact_5822_and__num_Opelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
=> ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(X,Xa2))
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_dd,bit_un7362597486090784418nd_num(M3,N2)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
tff(fact_5823_int_Obi__total,axiom,
bi_tot896582865486249351at_int(pcr_int) ).
% int.bi_total
tff(fact_5824_Domainp__cr__rat,axiom,
! [X3: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(domain9213661015745956397nt_rat(pcr_rat),X3))
<=> ( product_snd_int_int(X3) != zero_zero_int ) ) ).
% Domainp_cr_rat
tff(fact_5825_eventually__sequentially__Suc,axiom,
! [P: fun_nat_bool] :
( eventually_nat(aTP_Lamm_ge(P),at_top_nat)
<=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentially_Suc
tff(fact_5826_eventually__sequentially__seg,axiom,
! [P: fun_nat_bool,K: nat] :
( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_gg(P),K),at_top_nat)
<=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentially_seg
tff(fact_5827_eventually__sequentially,axiom,
! [P: fun_nat_bool] :
( eventually_nat(P,at_top_nat)
<=> ? [N8: nat] :
! [N3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N3))
=> pp(aa_nat_bool(P,N3)) ) ) ).
% eventually_sequentially
tff(fact_5828_eventually__sequentiallyI,axiom,
! [C: nat,P: fun_nat_bool] :
( ! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X2))
=> pp(aa_nat_bool(P,X2)) )
=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentiallyI
tff(fact_5829_le__sequentially,axiom,
! [F3: filter_nat] :
( ord_le2510731241096832064er_nat(F3,at_top_nat)
<=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),F3) ) ).
% le_sequentially
tff(fact_5830_eventually__False__sequentially,axiom,
~ eventually_nat(aTP_Lamm_cq,at_top_nat) ).
% eventually_False_sequentially
tff(fact_5831_int_Oright__total,axiom,
right_1157779049177982926at_int(pcr_int) ).
% int.right_total
tff(fact_5832_pair__lessI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
=> member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).
% pair_lessI2
tff(fact_5833_pair__leqI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T))
=> member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).
% pair_leqI2
tff(fact_5834_euclidean__size__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_nat(euclid6377331345833325938nteger,X) = aa_int_nat(euclid4774559944035922753ze_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% euclidean_size_integer.rep_eq
tff(fact_5835_euclidean__size__integer__def,axiom,
euclid6377331345833325938nteger = map_fu6539832666145259331at_nat(code_int_of_integer,id_nat,euclid4774559944035922753ze_int) ).
% euclidean_size_integer_def
tff(fact_5836_euclidean__size__integer_Oabs__eq,axiom,
! [X: int] : aa_Code_integer_nat(euclid6377331345833325938nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(euclid4774559944035922753ze_int,X) ).
% euclidean_size_integer.abs_eq
tff(fact_5837_euclidean__size__natural_Orsp,axiom,
pp(aa_fun_nat_nat_bool(aa_fun3185460345579526648t_bool(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),euclid4777050414544973029ze_nat),euclid4777050414544973029ze_nat)) ).
% euclidean_size_natural.rsp
tff(fact_5838_euclidean__size__integer_Orsp,axiom,
bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,euclid4774559944035922753ze_int,euclid4774559944035922753ze_int) ).
% euclidean_size_integer.rsp
tff(fact_5839_division__segment__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,X)) = aa_int_int(euclid3395696857347342551nt_int,aa_Code_integer_int(code_int_of_integer,X)) ).
% division_segment_integer.rep_eq
tff(fact_5840_abs__division__segment,axiom,
! [K: int] : aa_int_int(abs_abs_int,aa_int_int(euclid3395696857347342551nt_int,K)) = one_one_int ).
% abs_division_segment
tff(fact_5841_division__segment__integer_Orsp,axiom,
pp(aa_fun_int_int_bool(aa_fun3770596143487498376t_bool(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),euclid3395696857347342551nt_int),euclid3395696857347342551nt_int)) ).
% division_segment_integer.rsp
tff(fact_5842_division__segment__natural_Orsp,axiom,
pp(aa_fun_nat_nat_bool(aa_fun3185460345579526648t_bool(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),euclid3398187327856392827nt_nat),euclid3398187327856392827nt_nat)) ).
% division_segment_natural.rsp
tff(fact_5843_division__segment__nat__def,axiom,
! [N: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,N) = one_one_nat ).
% division_segment_nat_def
tff(fact_5844_division__segment__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(euclid3395696857347342551nt_int,X)) ).
% division_segment_integer.abs_eq
tff(fact_5845_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(sgn_sgn_int,K) ) ) ).
% division_segment_eq_sgn
tff(fact_5846_division__segment__integer__def,axiom,
euclid6289375185220004616nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),euclid3395696857347342551nt_int) ).
% division_segment_integer_def
tff(fact_5847_division__segment__int__def,axiom,
! [K: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( aa_int_int(euclid3395696857347342551nt_int,K) = one_one_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% division_segment_int_def
tff(fact_5848_wf__int__ge__less__than2,axiom,
! [D: int] : pp(aa_set7317184552576006522t_bool(wf_int,int_ge_less_than2(D))) ).
% wf_int_ge_less_than2
tff(fact_5849_wf__int__ge__less__than,axiom,
! [D: int] : pp(aa_set7317184552576006522t_bool(wf_int,int_ge_less_than(D))) ).
% wf_int_ge_less_than
tff(fact_5850_wf__less,axiom,
wf_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat))) ).
% wf_less
tff(fact_5851_num__of__integer_Otransfer,axiom,
bNF_re6718328864250387230um_num(code_pcr_integer,fequal_num,comp_nat_num_int(num_of_nat,nat2),code_num_of_integer) ).
% num_of_integer.transfer
tff(fact_5852_euclidean__size__integer_Otransfer,axiom,
bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,euclid4774559944035922753ze_int,euclid6377331345833325938nteger) ).
% euclidean_size_integer.transfer
tff(fact_5853_integer_Oleft__unique,axiom,
left_u6795595883297946906nteger(code_pcr_integer) ).
% integer.left_unique
tff(fact_5854_zero__integer_Otransfer,axiom,
pp(aa_Code_integer_bool(aa_int3865899974016033897r_bool(code_pcr_integer,zero_zero_int),zero_z3403309356797280102nteger)) ).
% zero_integer.transfer
tff(fact_5855_one__integer_Otransfer,axiom,
pp(aa_Code_integer_bool(aa_int3865899974016033897r_bool(code_pcr_integer,one_one_int),one_one_Code_integer)) ).
% one_integer.transfer
tff(fact_5856_integer_Obi__unique,axiom,
bi_uni8663861020805155098nteger(code_pcr_integer) ).
% integer.bi_unique
tff(fact_5857_integer_Odomain,axiom,
! [P: fun_int_bool] :
( ( domainp_int_int(fequal_int) = P )
=> ( domain452967906960104687nteger(code_pcr_integer) = P ) ) ).
% integer.domain
tff(fact_5858_integer_Obi__total,axiom,
bi_tot1331153423839324337nteger(code_pcr_integer) ).
% integer.bi_total
tff(fact_5859_flip__bit__integer_Otransfer,axiom,
bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se2159334234014336723it_int,bit_se1345352211410354436nteger) ).
% flip_bit_integer.transfer
tff(fact_5860_push__bit__integer_Otransfer,axiom,
bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se545348938243370406it_int,bit_se7788150548672797655nteger) ).
% push_bit_integer.transfer
tff(fact_5861_drop__bit__integer_Otransfer,axiom,
bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se8568078237143864401it_int,bit_se3928097537394005634nteger) ).
% drop_bit_integer.transfer
tff(fact_5862_xor__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se6526347334894502574or_int,bit_se3222712562003087583nteger) ).
% xor_integer.transfer
tff(fact_5863_bit__integer_Otransfer,axiom,
bNF_re4558401625797333864t_bool(code_pcr_integer,fequal_fun_nat_bool,bit_se1146084159140164899it_int,bit_se9216721137139052372nteger) ).
% bit_integer.transfer
tff(fact_5864_or__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se1409905431419307370or_int,bit_se1080825931792720795nteger) ).
% or_integer.transfer
tff(fact_5865_take__bit__integer_Otransfer,axiom,
bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se2923211474154528505it_int,bit_se1745604003318907178nteger) ).
% take_bit_integer.transfer
tff(fact_5866_not__integer_Otransfer,axiom,
pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_ri7919022796975470100ot_int),bit_ri7632146776885996613nteger)) ).
% not_integer.transfer
tff(fact_5867_and__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se725231765392027082nd_int,bit_se3949692690581998587nteger) ).
% and_integer.transfer
tff(fact_5868_dup_Otransfer,axiom,
pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),aTP_Lamm_fq),code_dup)) ).
% dup.transfer
tff(fact_5869_integer__of__nat_Otransfer,axiom,
bNF_re4153400068438556298nteger(fequal_nat,code_pcr_integer,semiri1314217659103216013at_int,code_integer_of_nat) ).
% integer_of_nat.transfer
tff(fact_5870_less__integer_Otransfer,axiom,
bNF_re6983685936609452663r_bool(code_pcr_integer,bNF_re3324582151780938536l_bool(code_pcr_integer,fequal_bool),ord_less_int,ord_le6747313008572928689nteger) ).
% less_integer.transfer
tff(fact_5871_times__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),times_times_int,times_3573771949741848930nteger) ).
% times_integer.transfer
tff(fact_5872_divide__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),divide_divide_int,divide6298287555418463151nteger) ).
% divide_integer.transfer
tff(fact_5873_abs__integer_Otransfer,axiom,
pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),abs_abs_int),abs_abs_Code_integer)) ).
% abs_integer.transfer
tff(fact_5874_nat__of__integer_Otransfer,axiom,
bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,nat2,code_nat_of_integer) ).
% nat_of_integer.transfer
tff(fact_5875_modulo__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),modulo_modulo_int,modulo364778990260209775nteger) ).
% modulo_integer.transfer
tff(fact_5876_plus__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),plus_plus_int,plus_p5714425477246183910nteger) ).
% plus_integer.transfer
tff(fact_5877_sgn__integer_Otransfer,axiom,
pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),sgn_sgn_int),sgn_sgn_Code_integer)) ).
% sgn_integer.transfer
tff(fact_5878_integer_Orep__transfer,axiom,
bNF_re3804157879324367682nt_int(code_pcr_integer,fequal_int,aTP_Lamm_bu,code_int_of_integer) ).
% integer.rep_transfer
tff(fact_5879_integer_Oid__abs__transfer,axiom,
bNF_re982302072995117890nteger(fequal_int,code_pcr_integer,aTP_Lamm_bu,code_integer_of_int) ).
% integer.id_abs_transfer
tff(fact_5880_gcd__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),gcd_gcd_int,gcd_gcd_Code_integer) ).
% gcd_integer.transfer
tff(fact_5881_minus__integer_Otransfer,axiom,
bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),minus_minus_int,minus_8373710615458151222nteger) ).
% minus_integer.transfer
tff(fact_5882_uminus__integer_Otransfer,axiom,
pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),uminus_uminus_int),uminus1351360451143612070nteger)) ).
% uminus_integer.transfer
tff(fact_5883_mask__integer_Otransfer,axiom,
bNF_re4153400068438556298nteger(fequal_nat,code_pcr_integer,bit_se2000444600071755411sk_int,bit_se2119862282449309892nteger) ).
% mask_integer.transfer
tff(fact_5884_unset__bit__integer_Otransfer,axiom,
bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se4203085406695923979it_int,bit_se8260200283734997820nteger) ).
% unset_bit_integer.transfer
tff(fact_5885_set__bit__integer_Otransfer,axiom,
bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se7879613467334960850it_int,bit_se2793503036327961859nteger) ).
% set_bit_integer.transfer
tff(fact_5886_integer_Oright__total,axiom,
right_7281159819619925802nteger(code_pcr_integer) ).
% integer.right_total
tff(fact_5887_less__eq__integer_Otransfer,axiom,
bNF_re6983685936609452663r_bool(code_pcr_integer,bNF_re3324582151780938536l_bool(code_pcr_integer,fequal_bool),ord_less_eq_int,ord_le3102999989581377725nteger) ).
% less_eq_integer.transfer
tff(fact_5888_division__segment__integer_Otransfer,axiom,
pp(aa_fun7169572220525660642r_bool(aa_fun4674266849241344294r_bool(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),euclid3395696857347342551nt_int),euclid6289375185220004616nteger)) ).
% division_segment_integer.transfer
tff(fact_5889_sub_Otransfer,axiom,
bNF_re4520166952630509772nteger(fequal_num,bNF_re6501075790457514782nteger(fequal_num,code_pcr_integer),aTP_Lamm_fr,code_sub) ).
% sub.transfer
tff(fact_5890_eventually__prod__sequentially,axiom,
! [P: fun_Pr7757163693017731481t_bool] :
( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
<=> ? [N8: nat] :
! [M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M5))
=> ! [N3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N3))
=> pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M5))) ) ) ) ).
% eventually_prod_sequentially
tff(fact_5891_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_5892_inj__on__char__of__nat,axiom,
pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).
% inj_on_char_of_nat
tff(fact_5893_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_5894_char_Osize_I2_J,axiom,
! [X12: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X7: bool,X8: bool] : size_size_char(aa_bool_char(char2(X12,X22,X32,X42,X52,X62,X7),X8)) = zero_zero_nat ).
% char.size(2)
tff(fact_5895_integer__of__char__code,axiom,
! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B5: bool,B6: bool,B7: bool] : integer_of_char(aa_bool_char(char2(B0,B1,B22,B32,B42,B5,B6),B7)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(B7)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B6))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B5))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B42))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B32))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B22))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B1))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),zero_n356916108424825756nteger(B0)) ).
% integer_of_char_code
tff(fact_5896_char_Osize__gen,axiom,
! [X12: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X7: bool,X8: bool] : size_char(aa_bool_char(char2(X12,X22,X32,X42,X52,X62,X7),X8)) = zero_zero_nat ).
% char.size_gen
tff(fact_5897_char__of__integer__code,axiom,
! [K: code_integer] : char_of_integer(K) = produc7257819798349884575l_char(aTP_Lamm_go,code_bit_cut_integer(K)) ).
% char_of_integer_code
tff(fact_5898_assert_H__def,axiom,
! [P: bool] :
( ( pp(P)
=> ( refine_Imp_assert(P) = aa_Pro1108561329362217960t_unit(heap_T4284346855313245393t_unit,product_Unity) ) )
& ( ~ pp(P)
=> ( refine_Imp_assert(P) = heap_T2927564422264180874t_unit(cons_char(aa_bool_char(char2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue),fFalse),cons_char(aa_bool_char(char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue),fFalse),cons_char(aa_bool_char(char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue),fFalse),cons_char(aa_bool_char(char2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue),fFalse),cons_char(aa_bool_char(char2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue),fFalse),cons_char(aa_bool_char(char2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue),fFalse),nil_char))))))) ) ) ) ).
% assert'_def
tff(fact_5899_wait__def,axiom,
! [N: nat] : heap_Time_wait(N) = heap_T6183433275982383450t_unit(aTP_Lamm_gp(N)) ).
% wait_def
tff(fact_5900_unit__all__impI,axiom,
! [P: fun_Pr1572506220757514509t_bool,Q: fun_Pr1572506220757514509t_bool] :
( ( pp(aa_Product_unit_bool(P,product_Unity))
=> pp(aa_Product_unit_bool(Q,product_Unity)) )
=> ! [X3: product_unit] :
( pp(aa_Product_unit_bool(P,X3))
=> pp(aa_Product_unit_bool(Q,X3)) ) ) ).
% unit_all_impI
tff(fact_5901_eq__sym__Unity__conv,axiom,
! [X: bool] :
( ( pp(X)
<=> ( product_Unity = product_Unity ) )
<=> pp(X) ) ).
% eq_sym_Unity_conv
tff(fact_5902_refines__assert_H,axiom,
! [Phi: bool] : refine451278484176762712t_unit(aa_Pro1108561329362217960t_unit(heap_T4284346855313245393t_unit,product_Unity),refine_Imp_assert(Phi)) ).
% refines_assert'
tff(fact_5903_execute__assert_H,axiom,
! [P: bool,H: heap_e7401611519738050253t_unit] :
( ( pp(P)
=> ( heap_T875086893843062177t_unit(refine_Imp_assert(P),H) = some_P1914260805536162275it_nat(produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(H,zero_zero_nat))) ) )
& ( ~ pp(P)
=> ( heap_T875086893843062177t_unit(refine_Imp_assert(P),H) = none_P9117596204409417319it_nat ) ) ) ).
% execute_assert'
tff(fact_5904_natural__zero__minus__one,axiom,
aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,zero_z2226904508553997617atural),one_one_Code_natural) = zero_z2226904508553997617atural ).
% natural_zero_minus_one
tff(fact_5905_minus__shift__def,axiom,
! [K: code_natural,L: code_natural,R: code_natural] :
( ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,K),L))
=> ( minus_shift(R,K,L) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,R),K)),L) ) )
& ( ~ pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,K),L))
=> ( minus_shift(R,K,L) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,K),L) ) ) ) ).
% minus_shift_def
tff(fact_5906_next_Osimps,axiom,
! [V: code_natural,W: code_natural] : aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W)) = aa_Pro7165575656943818355atural(produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,minus_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,V),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,V),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,W),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,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,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,W),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))),one_one_Code_natural))),one_one_Code_natural)),produc3574140220909816553atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,V),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,V),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,W),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,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,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,W),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))) ).
% next.simps
tff(fact_5907_log_Oelims,axiom,
! [X: code_natural,Xa2: code_natural,Y: code_natural] :
( ( log(X,Xa2) = Y )
=> ( ( ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,X),one_one_Code_natural))
| pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa2),X)) )
=> ( Y = one_one_Code_natural ) )
& ( ~ ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,X),one_one_Code_natural))
| pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa2),X)) )
=> ( Y = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,Xa2),X))) ) ) ) ) ).
% log.elims
tff(fact_5908_log_Osimps,axiom,
! [B: code_natural,I: code_natural] :
( ( ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),one_one_Code_natural))
| pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,I),B)) )
=> ( log(B,I) = one_one_Code_natural ) )
& ( ~ ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,B),one_one_Code_natural))
| pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,I),B)) )
=> ( log(B,I) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(B,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,I),B))) ) ) ) ).
% log.simps
tff(fact_5909_split__seed__def,axiom,
! [S: produc7822875418678951345atural] : split_seed(S) = produc8282080750456430313atural(aTP_Lamm_gr(S),S) ).
% split_seed_def
tff(fact_5910_Random_Orange__def,axiom,
! [K: code_natural] : range(K) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),K),aTP_Lamm_gt,one_one_Code_natural),aTP_Lamm_gu(K)) ).
% Random.range_def
tff(fact_5911_inc__shift__def,axiom,
! [V: code_natural,K: code_natural] :
( ( ( V = K )
=> ( inc_shift(V,K) = one_one_Code_natural ) )
& ( ( V != K )
=> ( inc_shift(V,K) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,K),one_one_Code_natural) ) ) ) ).
% inc_shift_def
tff(fact_5912_log_Opelims,axiom,
! [X: code_natural,Xa2: code_natural,Y: code_natural] :
( ( log(X,Xa2) = Y )
=> ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa2))
=> ~ ( ( ( ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,X),one_one_Code_natural))
| pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa2),X)) )
=> ( Y = one_one_Code_natural ) )
& ( ~ ( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,X),one_one_Code_natural))
| pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,Xa2),X)) )
=> ( Y = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,Xa2),X))) ) ) )
=> ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa2)) ) ) ) ).
% log.pelims
tff(fact_5913_natural__of__nat__of__nat,axiom,
code_natural_of_nat = semiri3763490453095760265atural ).
% natural_of_nat_of_nat
tff(fact_5914_divide__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa2),X)) ).
% divide_natural.abs_eq
tff(fact_5915_less__eq__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X)) ) ).
% less_eq_natural.abs_eq
tff(fact_5916_minus__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),X)) ).
% minus_natural.abs_eq
tff(fact_5917_zero__natural__def,axiom,
zero_z2226904508553997617atural = aa_nat_Code_natural(code_natural_of_nat,zero_zero_nat) ).
% zero_natural_def
tff(fact_5918_one__natural__def,axiom,
one_one_Code_natural = aa_nat_Code_natural(code_natural_of_nat,one_one_nat) ).
% one_natural_def
tff(fact_5919_euclidean__size__natural_Oabs__eq,axiom,
! [X: nat] : aa_Code_natural_nat(euclid5200926497590043453atural,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_nat(euclid4777050414544973029ze_nat,X) ).
% euclidean_size_natural.abs_eq
tff(fact_5920_mask__natural__def,axiom,
bit_se943457434206027407atural = map_fu4742413587575543474atural(id_nat,code_natural_of_nat,bit_se2002935070580805687sk_nat) ).
% mask_natural_def
tff(fact_5921_drop__bit__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se2751692689150723149atural,Xa2),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Xa2),X)) ).
% drop_bit_natural.abs_eq
tff(fact_5922_and__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(bit_se2773287842338716102atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Xa2),X)) ).
% and_natural.abs_eq
tff(fact_5923_bit__natural_Oabs__eq,axiom,
! [X: nat] : aa_Cod140358652193338554t_bool(bit_se8040316288895769887atural,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,X) ).
% bit_natural.abs_eq
tff(fact_5924_take__bit__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se569199155075624693atural,Xa2),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Xa2),X)) ).
% take_bit_natural.abs_eq
tff(fact_5925_xor__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(bit_se2046307713759805098atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Xa2),X)) ).
% xor_natural.abs_eq
tff(fact_5926_or__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(bit_se9127793120404214118atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Xa2),X)) ).
% or_natural.abs_eq
tff(fact_5927_push__bit__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se6611745700429515170atural,Xa2),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Xa2),X)) ).
% push_bit_natural.abs_eq
tff(fact_5928_mask__natural_Oabs__eq,axiom,
! [X: nat] : aa_nat_Code_natural(bit_se943457434206027407atural,X) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,X)) ).
% mask_natural.abs_eq
tff(fact_5929_flip__bit__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se168947363167071951atural,Xa2),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Xa2),X)) ).
% flip_bit_natural.abs_eq
tff(fact_5930_unset__bit__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se7083795435491715335atural,Xa2),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Xa2),X)) ).
% unset_bit_natural.abs_eq
tff(fact_5931_set__bit__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se1617098188084679374atural,Xa2),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Xa2),X)) ).
% set_bit_natural.abs_eq
tff(fact_5932_division__segment__natural_Oabs__eq,axiom,
! [X: nat] : aa_Cod4272041906116450446atural(euclid5112970336976722131atural,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(euclid3398187327856392827nt_nat,X)) ).
% division_segment_natural.abs_eq
tff(fact_5933_natural__of__nat__cases,axiom,
! [X: code_natural] :
~ ! [Y2: nat] :
( ( X = aa_nat_Code_natural(code_natural_of_nat,Y2) )
=> ~ pp(aa_set_nat_bool(member_nat(Y2),top_top_set_nat)) ) ).
% natural_of_nat_cases
tff(fact_5934_natural__of__nat__induct,axiom,
! [P: fun_Co7175331531449502024l_bool,X: code_natural] :
( ! [Y2: nat] :
( pp(aa_set_nat_bool(member_nat(Y2),top_top_set_nat))
=> pp(aa_Code_natural_bool(P,aa_nat_Code_natural(code_natural_of_nat,Y2))) )
=> pp(aa_Code_natural_bool(P,X)) ) ).
% natural_of_nat_induct
tff(fact_5935_natural__of__nat__inject,axiom,
! [X: nat,Y: nat] :
( pp(aa_set_nat_bool(member_nat(X),top_top_set_nat))
=> ( pp(aa_set_nat_bool(member_nat(Y),top_top_set_nat))
=> ( ( aa_nat_Code_natural(code_natural_of_nat,X) = aa_nat_Code_natural(code_natural_of_nat,Y) )
<=> ( X = Y ) ) ) ) ).
% natural_of_nat_inject
tff(fact_5936_less__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),X)) ) ).
% less_natural.abs_eq
tff(fact_5937_plus__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),X)) ).
% plus_natural.abs_eq
tff(fact_5938_times__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa2),X)) ).
% times_natural.abs_eq
tff(fact_5939_modulo__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa2),X)) ).
% modulo_natural.abs_eq
tff(fact_5940_Nat_Oabs__eq,axiom,
! [X: int] : aa_Cod607569804290114243atural(code_Nat,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_int_nat(nat2,X)) ).
% Nat.abs_eq
tff(fact_5941_nat__of__natural__inverse,axiom,
! [X: code_natural] : aa_nat_Code_natural(code_natural_of_nat,aa_Code_natural_nat(code_nat_of_natural,X)) = X ).
% nat_of_natural_inverse
tff(fact_5942_nat__of__natural__of__nat__inverse,axiom,
! [N: nat] : aa_Code_natural_nat(code_nat_of_natural,aa_nat_Code_natural(code_natural_of_nat,N)) = N ).
% nat_of_natural_of_nat_inverse
tff(fact_5943_natural__of__nat__of__natural__inverse,axiom,
! [N: code_natural] : aa_nat_Code_natural(code_natural_of_nat,aa_Code_natural_nat(code_nat_of_natural,N)) = N ).
% natural_of_nat_of_natural_inverse
tff(fact_5944_nat__of__natural__min,axiom,
! [K: code_natural,L: code_natural] : aa_Code_natural_nat(code_nat_of_natural,ord_min_Code_natural(K,L)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_Code_natural_nat(code_nat_of_natural,K)),aa_Code_natural_nat(code_nat_of_natural,L)) ).
% nat_of_natural_min
tff(fact_5945_nat__of__natural__max,axiom,
! [K: code_natural,L: code_natural] : aa_Code_natural_nat(code_nat_of_natural,ord_max_Code_natural(K,L)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_Code_natural_nat(code_nat_of_natural,K)),aa_Code_natural_nat(code_nat_of_natural,L)) ).
% nat_of_natural_max
tff(fact_5946_nat__of__natural__of__nat,axiom,
! [N: nat] : aa_Code_natural_nat(code_nat_of_natural,aa_nat_Code_natural(semiri3763490453095760265atural,N)) = N ).
% nat_of_natural_of_nat
tff(fact_5947_of__nat__of__natural,axiom,
! [N: code_natural] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_Code_natural_nat(code_nat_of_natural,N)) = N ).
% of_nat_of_natural
tff(fact_5948_zero__natural_Orep__eq,axiom,
aa_Code_natural_nat(code_nat_of_natural,zero_z2226904508553997617atural) = zero_zero_nat ).
% zero_natural.rep_eq
tff(fact_5949_one__natural_Orep__eq,axiom,
aa_Code_natural_nat(code_nat_of_natural,one_one_Code_natural) = one_one_nat ).
% one_natural.rep_eq
tff(fact_5950_plus__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% plus_natural.rep_eq
tff(fact_5951_minus__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% minus_natural.rep_eq
tff(fact_5952_nat__of__natural__numeral,axiom,
! [K: num] : aa_Code_natural_nat(code_nat_of_natural,numera5444537566228673987atural(K)) = aa_num_nat(numeral_numeral_nat,K) ).
% nat_of_natural_numeral
tff(fact_5953_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% times_natural.rep_eq
tff(fact_5954_divide__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(divide5121882707175180666atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% divide_natural.rep_eq
tff(fact_5955_modulo__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% modulo_natural.rep_eq
tff(fact_5956_division__segment__natural_Orep__eq,axiom,
! [X: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(euclid5112970336976722131atural,X)) = aa_nat_nat(euclid3398187327856392827nt_nat,aa_Code_natural_nat(code_nat_of_natural,X)) ).
% division_segment_natural.rep_eq
tff(fact_5957_euclidean__size__natural_Orep__eq,axiom,
! [X: code_natural] : aa_Code_natural_nat(euclid5200926497590043453atural,X) = aa_nat_nat(euclid4777050414544973029ze_nat,aa_Code_natural_nat(code_nat_of_natural,X)) ).
% euclidean_size_natural.rep_eq
tff(fact_5958_minus__natural__def,axiom,
minus_7197305767214868737atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),minus_minus_nat) ).
% minus_natural_def
tff(fact_5959_or__natural__def,axiom,
bit_se9127793120404214118atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se1412395901928357646or_nat) ).
% or_natural_def
tff(fact_5960_xor__natural__def,axiom,
bit_se2046307713759805098atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se6528837805403552850or_nat) ).
% xor_natural_def
tff(fact_5961_and__natural__def,axiom,
bit_se2773287842338716102atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se727722235901077358nd_nat) ).
% and_natural_def
tff(fact_5962_division__segment__natural__def,axiom,
euclid5112970336976722131atural = aa_fun5114738747585492520atural(map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),euclid3398187327856392827nt_nat) ).
% division_segment_natural_def
tff(fact_5963_natural__of__nat__inverse,axiom,
! [Y: nat] :
( pp(aa_set_nat_bool(member_nat(Y),top_top_set_nat))
=> ( aa_Code_natural_nat(code_nat_of_natural,aa_nat_Code_natural(code_natural_of_nat,Y)) = Y ) ) ).
% natural_of_nat_inverse
tff(fact_5964_type__definition__natural,axiom,
pp(aa_set_nat_bool(type_d4410041424927559462al_nat(code_nat_of_natural,code_natural_of_nat),top_top_set_nat)) ).
% type_definition_natural
tff(fact_5965_plus__natural__def,axiom,
plus_p4538020629002901425atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),plus_plus_nat) ).
% plus_natural_def
tff(fact_5966_less__natural__def,axiom,
ord_le5570908160329646204atural = map_fu675752389954593592l_bool(code_nat_of_natural,map_fu3095037182338179920l_bool(code_nat_of_natural,id_bool),ord_less_nat) ).
% less_natural_def
tff(fact_5967_less__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le5570908160329646204atural,X),Xa2))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2))) ) ).
% less_natural.rep_eq
tff(fact_5968_nat__of__natural__induct,axiom,
! [Y: nat,P: fun_nat_bool] :
( pp(aa_set_nat_bool(member_nat(Y),top_top_set_nat))
=> ( ! [X2: code_natural] : pp(aa_nat_bool(P,aa_Code_natural_nat(code_nat_of_natural,X2)))
=> pp(aa_nat_bool(P,Y)) ) ) ).
% nat_of_natural_induct
tff(fact_5969_nat__of__natural__cases,axiom,
! [Y: nat] :
( pp(aa_set_nat_bool(member_nat(Y),top_top_set_nat))
=> ~ ! [X2: code_natural] : Y != aa_Code_natural_nat(code_nat_of_natural,X2) ) ).
% nat_of_natural_cases
tff(fact_5970_nat__of__natural,axiom,
! [X: code_natural] : pp(aa_set_nat_bool(member_nat(aa_Code_natural_nat(code_nat_of_natural,X)),top_top_set_nat)) ).
% nat_of_natural
tff(fact_5971_drop__bit__natural_Orep__eq,axiom,
! [X: nat,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se2751692689150723149atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,X),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% drop_bit_natural.rep_eq
tff(fact_5972_bit__natural_Orep__eq,axiom,
! [X: code_natural] : aa_Cod140358652193338554t_bool(bit_se8040316288895769887atural,X) = aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_Code_natural_nat(code_nat_of_natural,X)) ).
% bit_natural.rep_eq
tff(fact_5973_or__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(bit_se9127793120404214118atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% or_natural.rep_eq
tff(fact_5974_xor__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(bit_se2046307713759805098atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% xor_natural.rep_eq
tff(fact_5975_natural__eq__iff,axiom,
! [M: code_natural,N: code_natural] :
( ( M = N )
<=> ( aa_Code_natural_nat(code_nat_of_natural,M) = aa_Code_natural_nat(code_nat_of_natural,N) ) ) ).
% natural_eq_iff
tff(fact_5976_natural__eqI,axiom,
! [M: code_natural,N: code_natural] :
( ( aa_Code_natural_nat(code_nat_of_natural,M) = aa_Code_natural_nat(code_nat_of_natural,N) )
=> ( M = N ) ) ).
% natural_eqI
tff(fact_5977_nat__of__natural__inject,axiom,
! [X: code_natural,Y: code_natural] :
( ( aa_Code_natural_nat(code_nat_of_natural,X) = aa_Code_natural_nat(code_nat_of_natural,Y) )
<=> ( X = Y ) ) ).
% nat_of_natural_inject
tff(fact_5978_take__bit__natural_Orep__eq,axiom,
! [X: nat,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se569199155075624693atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,X),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% take_bit_natural.rep_eq
tff(fact_5979_and__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(bit_se2773287842338716102atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% and_natural.rep_eq
tff(fact_5980_mask__natural_Orep__eq,axiom,
! [X: nat] : aa_Code_natural_nat(code_nat_of_natural,aa_nat_Code_natural(bit_se943457434206027407atural,X)) = aa_nat_nat(bit_se2002935070580805687sk_nat,X) ).
% mask_natural.rep_eq
tff(fact_5981_push__bit__natural_Orep__eq,axiom,
! [X: nat,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se6611745700429515170atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,X),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% push_bit_natural.rep_eq
tff(fact_5982_set__bit__natural_Orep__eq,axiom,
! [X: nat,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se1617098188084679374atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,X),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% set_bit_natural.rep_eq
tff(fact_5983_unset__bit__natural_Orep__eq,axiom,
! [X: nat,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se7083795435491715335atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,X),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% unset_bit_natural.rep_eq
tff(fact_5984_flip__bit__natural_Orep__eq,axiom,
! [X: nat,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_nat5282697797315388251atural(bit_se168947363167071951atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,X),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).
% flip_bit_natural.rep_eq
tff(fact_5985_Nat_Orep__eq,axiom,
! [X: code_integer] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod607569804290114243atural(code_Nat,X)) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).
% Nat.rep_eq
tff(fact_5986_euclidean__size__natural__def,axiom,
euclid5200926497590043453atural = map_fu226235371393106610at_nat(code_nat_of_natural,id_nat,euclid4777050414544973029ze_nat) ).
% euclidean_size_natural_def
tff(fact_5987_bit__natural__def,axiom,
bit_se8040316288895769887atural = map_fu2099429406741865360t_bool(code_nat_of_natural,id_fun_nat_bool,bit_se1148574629649215175it_nat) ).
% bit_natural_def
tff(fact_5988_less__eq__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] :
( pp(aa_Code_natural_bool(aa_Cod6910080592713026658l_bool(ord_le1926595141338095240atural,X),Xa2))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2))) ) ).
% less_eq_natural.rep_eq
tff(fact_5989_times__natural__def,axiom,
times_2397367101498566445atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),times_times_nat) ).
% times_natural_def
tff(fact_5990_divide__natural__def,axiom,
divide5121882707175180666atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),divide_divide_nat) ).
% divide_natural_def
tff(fact_5991_take__bit__natural__def,axiom,
bit_se569199155075624693atural = map_fu5588224966615042724atural(id_nat,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se2925701944663578781it_nat) ).
% take_bit_natural_def
tff(fact_5992_push__bit__natural__def,axiom,
bit_se6611745700429515170atural = map_fu5588224966615042724atural(id_nat,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se547839408752420682it_nat) ).
% push_bit_natural_def
tff(fact_5993_less__eq__natural__def,axiom,
ord_le1926595141338095240atural = map_fu675752389954593592l_bool(code_nat_of_natural,map_fu3095037182338179920l_bool(code_nat_of_natural,id_bool),ord_less_eq_nat) ).
% less_eq_natural_def
tff(fact_5994_modulo__natural__def,axiom,
modulo8411746178871703098atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),modulo_modulo_nat) ).
% modulo_natural_def
tff(fact_5995_drop__bit__natural__def,axiom,
bit_se2751692689150723149atural = map_fu5588224966615042724atural(id_nat,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se8570568707652914677it_nat) ).
% drop_bit_natural_def
tff(fact_5996_flip__bit__natural__def,axiom,
bit_se168947363167071951atural = map_fu5588224966615042724atural(id_nat,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se2161824704523386999it_nat) ).
% flip_bit_natural_def
tff(fact_5997_set__bit__natural__def,axiom,
bit_se1617098188084679374atural = map_fu5588224966615042724atural(id_nat,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se7882103937844011126it_nat) ).
% set_bit_natural_def
tff(fact_5998_unset__bit__natural__def,axiom,
bit_se7083795435491715335atural = map_fu5588224966615042724atural(id_nat,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),bit_se4205575877204974255it_nat) ).
% unset_bit_natural_def
tff(fact_5999_natural__decr,axiom,
! [N: code_natural] :
( ( N != zero_z2226904508553997617atural )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,N)),aa_nat_nat(suc,zero_zero_nat))),aa_Code_natural_nat(code_nat_of_natural,N))) ) ).
% natural_decr
tff(fact_6000_Nat__def,axiom,
code_Nat = map_fu1051355602067684763atural(code_int_of_integer,code_natural_of_nat,nat2) ).
% Nat_def
tff(fact_6001_integer__of__natural_Orep__eq,axiom,
! [X: code_natural] : aa_Code_integer_int(code_int_of_integer,aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,X)) = aa_nat_int(semiri1314217659103216013at_int,aa_Code_natural_nat(code_nat_of_natural,X)) ).
% integer_of_natural.rep_eq
tff(fact_6002_int__of__integer__of__natural,axiom,
! [N: code_natural] : aa_Code_integer_int(code_int_of_integer,aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_Code_natural_nat(code_nat_of_natural,N)) ).
% int_of_integer_of_natural
tff(fact_6003_integer__of__natural__of__nat,axiom,
! [N: nat] : aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,aa_nat_Code_natural(semiri3763490453095760265atural,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).
% integer_of_natural_of_nat
tff(fact_6004_integer__of__natural__def,axiom,
code_i5400310926305786745atural = map_fu2787874002554666395nteger(code_nat_of_natural,code_integer_of_int,semiri1314217659103216013at_int) ).
% integer_of_natural_def
tff(fact_6005_integer__of__natural_Oabs__eq,axiom,
! [X: nat] : aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,X)) ).
% integer_of_natural.abs_eq
tff(fact_6006_integer__of__natural__of__integer,axiom,
! [K: code_integer] : aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,aa_Cod607569804290114243atural(code_n4118661773612635043nteger,K)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),K) ).
% integer_of_natural_of_integer
tff(fact_6007_natural__of__integer__def,axiom,
code_n4118661773612635043nteger = map_fu1051355602067684763atural(code_int_of_integer,code_natural_of_nat,nat2) ).
% natural_of_integer_def
tff(fact_6008_natural__of__integer__of__natural,axiom,
! [N: code_natural] : aa_Cod607569804290114243atural(code_n4118661773612635043nteger,aa_Cod5448446754359732931nteger(code_i5400310926305786745atural,N)) = N ).
% natural_of_integer_of_natural
tff(fact_6009_natural__of__integer_Orep__eq,axiom,
! [X: code_integer] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod607569804290114243atural(code_n4118661773612635043nteger,X)) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).
% natural_of_integer.rep_eq
tff(fact_6010_natural__of__integer_Oabs__eq,axiom,
! [X: int] : aa_Cod607569804290114243atural(code_n4118661773612635043nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_int_nat(nat2,X)) ).
% natural_of_integer.abs_eq
tff(fact_6011_Code__Numeral_Osize__nat__def,axiom,
code_size_nat = code_nat_of_natural ).
% Code_Numeral.size_nat_def
tff(fact_6012_Code__Numeral_OSuc__def,axiom,
code_Suc = aa_fun5114738747585492520atural(map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),suc) ).
% Code_Numeral.Suc_def
tff(fact_6013_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( aa_Cod4272041906116450446atural(code_Suc,Natural) = aa_Cod4272041906116450446atural(code_Suc,Natural2) )
<=> ( Natural = Natural2 ) ) ).
% natural.inject
tff(fact_6014_Suc_Orep__eq,axiom,
! [X: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(code_Suc,X)) = aa_nat_nat(suc,aa_Code_natural_nat(code_nat_of_natural,X)) ).
% Suc.rep_eq
tff(fact_6015_natural_Odistinct_I2_J,axiom,
! [Natural2: code_natural] : aa_Cod4272041906116450446atural(code_Suc,Natural2) != zero_z2226904508553997617atural ).
% natural.distinct(2)
tff(fact_6016_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] : zero_z2226904508553997617atural != aa_Cod4272041906116450446atural(code_Suc,Natural2) ).
% natural.distinct(1)
tff(fact_6017_natural_Oexhaust,axiom,
! [Y: code_natural] :
( ( Y != zero_z2226904508553997617atural )
=> ~ ! [Natural3: code_natural] : Y != aa_Cod4272041906116450446atural(code_Suc,Natural3) ) ).
% natural.exhaust
tff(fact_6018_Suc_Oabs__eq,axiom,
! [X: nat] : aa_Cod4272041906116450446atural(code_Suc,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(suc,X)) ).
% Suc.abs_eq
tff(fact_6019_Suc__natural__minus__one,axiom,
! [N: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(minus_7197305767214868737atural,aa_Cod4272041906116450446atural(code_Suc,N)),one_one_Code_natural) = N ).
% Suc_natural_minus_one
tff(fact_6020_prod__decode__triangle__add,axiom,
! [K: nat,M: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) ).
% prod_decode_triangle_add
tff(fact_6021_prod__decode__def,axiom,
nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).
% prod_decode_def
tff(fact_6022_list__decode_Oelims,axiom,
! [X: nat,Y: list_nat] :
( ( nat_list_decode(X) = Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != nil_nat ) )
=> ~ ! [N2: nat] :
( ( X = aa_nat_nat(suc,N2) )
=> ( Y != produc2761476792215241774st_nat(aTP_Lamm_gv,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).
% list_decode.elims
tff(fact_6023_list__decode_Osimps_I2_J,axiom,
! [N: nat] : nat_list_decode(aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_gv,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).
% list_decode.simps(2)
tff(fact_6024_list__decode_Osimps_I1_J,axiom,
nat_list_decode(zero_zero_nat) = nil_nat ).
% list_decode.simps(1)
tff(fact_6025_list__decode_Opelims,axiom,
! [X: nat,Y: list_nat] :
( ( nat_list_decode(X) = Y )
=> ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),X))
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = nil_nat )
=> ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat)) ) )
=> ~ ! [N2: nat] :
( ( X = aa_nat_nat(suc,N2) )
=> ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_gv,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
=> ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))) ) ) ) ) ) ).
% list_decode.pelims
tff(fact_6026_list__decode_Opsimps_I1_J,axiom,
( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
=> ( nat_list_decode(zero_zero_nat) = nil_nat ) ) ).
% list_decode.psimps(1)
tff(fact_6027_list__decode_Opinduct,axiom,
! [A0: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),A0))
=> ( ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
=> pp(aa_nat_bool(P,zero_zero_nat)) )
=> ( ! [N2: nat] :
( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)))
=> ( ! [X3: nat,Y3: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) = aa_nat7603221163087113075at_nat(nat_prod_decode,N2) )
=> pp(aa_nat_bool(P,Y3)) )
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
=> pp(aa_nat_bool(P,A0)) ) ) ) ).
% list_decode.pinduct
tff(fact_6028_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)))
=> ( nat_list_decode(aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_gv,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ).
% list_decode.psimps(2)
tff(fact_6029_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),aa_num_num(bit0,N)) = aa_num_Code_integer(code_Neg,aa_num_num(bitM,N)) ).
% Code_Numeral.sub_code(4)
tff(fact_6030_less__eq__integer__code_I9_J,axiom,
! [K: num,L: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(code_Neg,K)),aa_num_Code_integer(code_Neg,L)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),K)) ) ).
% less_eq_integer_code(9)
tff(fact_6031_less__eq__integer__code_I3_J,axiom,
! [L: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(code_Neg,L))) ).
% less_eq_integer_code(3)
tff(fact_6032_less__eq__integer__code_I7_J,axiom,
! [K: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(code_Neg,K)),zero_z3403309356797280102nteger)) ).
% less_eq_integer_code(7)
tff(fact_6033_minus__integer__code_I6_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(code_Neg,M)),aa_num_Code_integer(code_Neg,N)) = aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,N),M) ).
% minus_integer_code(6)
tff(fact_6034_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(code_Neg,N)) = aa_num_Code_integer(code_Neg,aa_num_num(bit0,N)) ).
% Code_Numeral.dup_code(3)
tff(fact_6035_less__integer__code_I9_J,axiom,
! [K: num,L: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(code_Neg,K)),aa_num_Code_integer(code_Neg,L)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,L),K)) ) ).
% less_integer_code(9)
tff(fact_6036_plus__integer__code_I6_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(code_Neg,M)),aa_num_Code_integer(code_Neg,N)) = aa_num_Code_integer(code_Neg,aa_num_num(plus_plus_num(M),N)) ).
% plus_integer_code(6)
tff(fact_6037_less__integer__code_I7_J,axiom,
! [K: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(code_Neg,K)),zero_z3403309356797280102nteger)) ).
% less_integer_code(7)
tff(fact_6038_less__integer__code_I3_J,axiom,
! [L: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(code_Neg,L))) ).
% less_integer_code(3)
tff(fact_6039_divmod__abs__code_I2_J,axiom,
! [K: num,L: num] : code_divmod_abs(aa_num_Code_integer(code_Neg,K),aa_num_Code_integer(code_Neg,L)) = unique3479559517661332726nteger(K,L) ).
% divmod_abs_code(2)
tff(fact_6040_Code__Numeral_Osub__code_I5_J,axiom,
! [N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),aa_num_num(bit1,N)) = aa_num_Code_integer(code_Neg,aa_num_num(bit0,N)) ).
% Code_Numeral.sub_code(5)
tff(fact_6041_Gcd__int__set__eq__fold,axiom,
! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = fold_int_int(gcd_gcd_int,Xs,zero_zero_int) ).
% Gcd_int_set_eq_fold
tff(fact_6042_int_Oid__abs__transfer,axiom,
bNF_re7934895593101944656at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),pcr_int,aTP_Lamm_gw,abs_Integ) ).
% int.id_abs_transfer
tff(fact_6043_rat_Odomain,axiom,
! [X3: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(domain9213661015745956397nt_rat(pcr_rat),X3))
<=> ? [Y5: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(basic_4387203522000727145nt_int(fequal_int,fequal_int),X3),Y5))
& pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Y5),Y5)) ) ) ).
% rat.domain
tff(fact_6044_rat_Odomain__par,axiom,
! [DR1: fun_int_bool,DR2: fun_int_bool,P22: fun_Pr803113274115125473t_bool] :
( ( domainp_int_int(fequal_int) = DR1 )
=> ( ( domainp_int_int(fequal_int) = DR2 )
=> ( bNF_re1394333895916556125l_bool(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_bool,P22,aTP_Lamm_gx)
=> ( domain9213661015745956397nt_rat(pcr_rat) = inf_in4513820918327576947t_bool(basic_1567116559311922317nt_int(DR1,DR2),P22) ) ) ) ) ).
% rat.domain_par
tff(fact_6045_rat_Odomain__par__left__total,axiom,
! [P2: fun_Pr803113274115125473t_bool] :
( left_t3131394472396969446nt_int(basic_4387203522000727145nt_int(fequal_int,fequal_int))
=> ( bNF_re1394333895916556125l_bool(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_bool,P2,aTP_Lamm_gx)
=> ( domain9213661015745956397nt_rat(pcr_rat) = P2 ) ) ) ).
% rat.domain_par_left_total
tff(fact_6046_Gcd__nat__set__eq__fold,axiom,
! [Xs: list_nat] : gcd_Gcd_nat(set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).
% Gcd_nat_set_eq_fold
tff(fact_6047_Abs__rat__cases,axiom,
! [X: rat] :
~ ! [Y2: set_Pr958786334691620121nt_int] :
( ( X = aa_set4118085860027418549nt_rat(abs_rat,Y2) )
=> ~ member2340774599025711042nt_int(Y2,collec5210948495886036740nt_int(aTP_Lamm_gy)) ) ).
% Abs_rat_cases
tff(fact_6048_Abs__rat__induct,axiom,
! [P: fun_rat_bool,X: rat] :
( ! [Y2: set_Pr958786334691620121nt_int] :
( member2340774599025711042nt_int(Y2,collec5210948495886036740nt_int(aTP_Lamm_gy))
=> pp(aa_rat_bool(P,aa_set4118085860027418549nt_rat(abs_rat,Y2))) )
=> pp(aa_rat_bool(P,X)) ) ).
% Abs_rat_induct
tff(fact_6049_int_Odomain,axiom,
! [DR1: fun_nat_bool,DR2: fun_nat_bool] :
( ( domainp_nat_nat(fequal_nat) = DR1 )
=> ( ( domainp_nat_nat(fequal_nat) = DR2 )
=> ( domain7287725470393535433at_int(pcr_int) = basic_570253317919918037at_nat(DR1,DR2) ) ) ) ).
% int.domain
tff(fact_6050_Abs__rat__inject,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( member2340774599025711042nt_int(X,collec5210948495886036740nt_int(aTP_Lamm_gy))
=> ( member2340774599025711042nt_int(Y,collec5210948495886036740nt_int(aTP_Lamm_gy))
=> ( ( aa_set4118085860027418549nt_rat(abs_rat,X) = aa_set4118085860027418549nt_rat(abs_rat,Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_rat_inject
tff(fact_6051_Abs__rat__inverse,axiom,
! [Y: set_Pr958786334691620121nt_int] :
( member2340774599025711042nt_int(Y,collec5210948495886036740nt_int(aTP_Lamm_gy))
=> ( aa_rat8624586542854104963nt_int(rep_rat,aa_set4118085860027418549nt_rat(abs_rat,Y)) = Y ) ) ).
% Abs_rat_inverse
tff(fact_6052_type__definition__rat,axiom,
type_d8554052265237484179nt_int(rep_rat,abs_rat,collec5210948495886036740nt_int(aTP_Lamm_gy)) ).
% type_definition_rat
tff(fact_6053_Rep__rat,axiom,
! [X: rat] : member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(rep_rat,X),collec5210948495886036740nt_int(aTP_Lamm_gy)) ).
% Rep_rat
tff(fact_6054_Rep__rat__cases,axiom,
! [Y: set_Pr958786334691620121nt_int] :
( member2340774599025711042nt_int(Y,collec5210948495886036740nt_int(aTP_Lamm_gy))
=> ~ ! [X2: rat] : Y != aa_rat8624586542854104963nt_int(rep_rat,X2) ) ).
% Rep_rat_cases
tff(fact_6055_Rep__rat__induct,axiom,
! [Y: set_Pr958786334691620121nt_int,P: fun_se3032438776331272065t_bool] :
( member2340774599025711042nt_int(Y,collec5210948495886036740nt_int(aTP_Lamm_gy))
=> ( ! [X2: rat] : pp(aa_set7317184552576006522t_bool(P,aa_rat8624586542854104963nt_int(rep_rat,X2)))
=> pp(aa_set7317184552576006522t_bool(P,Y)) ) ) ).
% Rep_rat_induct
tff(fact_6056_pcr__int__def,axiom,
pcr_int = relcom8911704206367525917at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),cr_int) ).
% pcr_int_def
tff(fact_6057_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
tff(fact_6058_cr__int__def,axiom,
! [X3: product_prod_nat_nat] : aa_Pro2928840338864870335t_bool(cr_int,X3) = aa_int_fun_int_bool(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X3)) ).
% cr_int_def
tff(fact_6059_Quotient__int,axiom,
quotie1194848508323700631at_int(intrel,abs_Integ,rep_Integ,cr_int) ).
% Quotient_int
tff(fact_6060_lcm__code__integer,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(gcd_lcm_Code_integer(A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,A),B)) ).
% lcm_code_integer
tff(fact_6061_lcm__0__iff__nat,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% lcm_0_iff_nat
tff(fact_6062_lcm__0__iff__int,axiom,
! [M: int,N: int] :
( ( gcd_lcm_int(M,N) = zero_zero_int )
<=> ( ( M = zero_zero_int )
| ( N = zero_zero_int ) ) ) ).
% lcm_0_iff_int
tff(fact_6063_lcm__proj2__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,X),Y) = Y ) ) ).
% lcm_proj2_if_dvd_nat
tff(fact_6064_lcm__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Y),X) = Y ) ) ).
% lcm_proj1_if_dvd_nat
tff(fact_6065_lcm__proj2__iff__nat,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = N )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).
% lcm_proj2_iff_nat
tff(fact_6066_lcm__proj1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = M )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).
% lcm_proj1_iff_nat
tff(fact_6067_lcm__int__int__eq,axiom,
! [M: nat,N: nat] : gcd_lcm_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ).
% lcm_int_int_eq
tff(fact_6068_abs__lcm__int,axiom,
! [I: int,J: int] : aa_int_int(abs_abs_int,gcd_lcm_int(I,J)) = gcd_lcm_int(I,J) ).
% abs_lcm_int
tff(fact_6069_lcm__abs1__int,axiom,
! [X: int,Y: int] : gcd_lcm_int(aa_int_int(abs_abs_int,X),Y) = gcd_lcm_int(X,Y) ).
% lcm_abs1_int
tff(fact_6070_lcm__abs2__int,axiom,
! [X: int,Y: int] : gcd_lcm_int(X,aa_int_int(abs_abs_int,Y)) = gcd_lcm_int(X,Y) ).
% lcm_abs2_int
tff(fact_6071_ATP_Olambda__1,axiom,
! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_fm,Uu) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(Uu)),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(Uu)),product_fst_int_int(Uu))) ).
% ATP.lambda_1
tff(fact_6072_ATP_Olambda__2,axiom,
! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_fo,Uu) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,product_fst_int_int(Uu))),product_snd_int_int(Uu)) ).
% ATP.lambda_2
tff(fact_6073_ATP_Olambda__3,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_fc,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).
% ATP.lambda_3
tff(fact_6074_ATP_Olambda__4,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_fq,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).
% ATP.lambda_4
tff(fact_6075_ATP_Olambda__5,axiom,
! [Uu: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_gx,Uu))
<=> pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Uu),Uu)) ) ).
% ATP.lambda_5
tff(fact_6076_ATP_Olambda__6,axiom,
! [Uu: code_natural] : aa_Cod4272041906116450446atural(aTP_Lamm_as,Uu) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uu),one_one_Code_natural) ).
% ATP.lambda_6
tff(fact_6077_ATP_Olambda__7,axiom,
! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_aq,Uu) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),one_one_Code_integer) ).
% ATP.lambda_7
tff(fact_6078_ATP_Olambda__8,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_ar,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).
% ATP.lambda_8
tff(fact_6079_ATP_Olambda__9,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_ap,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).
% ATP.lambda_9
tff(fact_6080_ATP_Olambda__10,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_ao,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).
% ATP.lambda_10
tff(fact_6081_ATP_Olambda__11,axiom,
! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_gb,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).
% ATP.lambda_11
tff(fact_6082_ATP_Olambda__12,axiom,
! [Uu: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_fk,Uu))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu)))) ) ).
% ATP.lambda_12
tff(fact_6083_ATP_Olambda__13,axiom,
! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_gt,Uu) = produc5538323210962509403atural(next,aTP_Lamm_gs(Uu)) ).
% ATP.lambda_13
tff(fact_6084_ATP_Olambda__14,axiom,
! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_ft,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cs,Uu)) ).
% ATP.lambda_14
tff(fact_6085_ATP_Olambda__15,axiom,
! [Uu: num] : aa_num_option_num(aTP_Lamm_dd,Uu) = some_num(aa_num_num(bit1,Uu)) ).
% ATP.lambda_15
tff(fact_6086_ATP_Olambda__16,axiom,
! [Uu: num] : aa_num_option_num(aTP_Lamm_di,Uu) = some_num(aa_num_num(bit0,Uu)) ).
% ATP.lambda_16
tff(fact_6087_ATP_Olambda__17,axiom,
! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_eq,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).
% ATP.lambda_17
tff(fact_6088_ATP_Olambda__18,axiom,
! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_er,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).
% ATP.lambda_18
tff(fact_6089_ATP_Olambda__19,axiom,
! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_dw,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).
% ATP.lambda_19
tff(fact_6090_ATP_Olambda__20,axiom,
! [Uu: int] : aa_int_nat(aTP_Lamm_de,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).
% ATP.lambda_20
tff(fact_6091_ATP_Olambda__21,axiom,
! [Uu: set_Pr958786334691620121nt_int] :
( pp(aa_set7317184552576006522t_bool(aTP_Lamm_gy,Uu))
<=> ? [X6: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X6),X6))
& ( Uu = collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(ratrel,X6)) ) ) ) ).
% ATP.lambda_21
tff(fact_6092_ATP_Olambda__22,axiom,
! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_du(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_ds(Uua),aTP_Lamm_dt(Uua),Uu) ).
% ATP.lambda_22
tff(fact_6093_ATP_Olambda__23,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_dl,Uu),Uua) = if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu))),one_one_int)) ).
% ATP.lambda_23
tff(fact_6094_ATP_Olambda__24,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_dk,Uu),Uua) = aa_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_num_num(plus_plus_num(aa_Code_integer_num(code_num_of_integer,Uu)),aa_Code_integer_num(code_num_of_integer,Uu))),aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(aa_Code_integer_num(code_num_of_integer,Uu)),aa_Code_integer_num(code_num_of_integer,Uu))),one)) ).
% ATP.lambda_24
tff(fact_6095_ATP_Olambda__25,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_dj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),one_one_nat)) ).
% ATP.lambda_25
tff(fact_6096_ATP_Olambda__26,axiom,
! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fa,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).
% ATP.lambda_26
tff(fact_6097_ATP_Olambda__27,axiom,
! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fn,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uua)) ).
% ATP.lambda_27
tff(fact_6098_ATP_Olambda__28,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_dv,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_du(Uua),Uu) ).
% ATP.lambda_28
tff(fact_6099_ATP_Olambda__29,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ds(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_di,bit_take_bit_num(Uu,Uua)) ).
% ATP.lambda_29
tff(fact_6100_ATP_Olambda__30,axiom,
! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_fl,Uu),Uua) = aa_int7462565939943104719nt_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,product_fst_int_int(Uu)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).
% ATP.lambda_30
tff(fact_6101_ATP_Olambda__31,axiom,
! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_fp,Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_fst_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).
% ATP.lambda_31
tff(fact_6102_ATP_Olambda__32,axiom,
! [Uu: rat,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_fg(Uu),Uua))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).
% ATP.lambda_32
tff(fact_6103_ATP_Olambda__33,axiom,
! [Uu: rat,Uua: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ga(Uu),Uua))
<=> ( ( Uu = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Uua)))
& pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua))) ) ) ).
% ATP.lambda_33
tff(fact_6104_ATP_Olambda__34,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cd,Uu),Uua))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
& ( Uu != Uua ) ) ) ).
% ATP.lambda_34
tff(fact_6105_ATP_Olambda__35,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% ATP.lambda_35
tff(fact_6106_ATP_Olambda__36,axiom,
! [Uu: set_int,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_gf(Uu),Uua))
<=> ( member_int(Uua,Uu)
& ! [X6: int] :
( member_int(X6,Uu)
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),Uua)) ) ) ) ).
% ATP.lambda_36
tff(fact_6107_ATP_Olambda__37,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).
% ATP.lambda_37
tff(fact_6108_ATP_Olambda__38,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ch,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).
% ATP.lambda_38
tff(fact_6109_ATP_Olambda__39,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).
% ATP.lambda_39
tff(fact_6110_ATP_Olambda__40,axiom,
! [Uu: code_integer,Uua: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aTP_Lamm_go,Uu),Uua) = produc7257819798349884575l_char(aTP_Lamm_gn(Uua),code_bit_cut_integer(Uu)) ).
% ATP.lambda_40
tff(fact_6111_ATP_Olambda__41,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).
% ATP.lambda_41
tff(fact_6112_ATP_Olambda__42,axiom,
! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_dc(Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).
% ATP.lambda_42
tff(fact_6113_ATP_Olambda__43,axiom,
! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_fr,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu)),aa_num_int(numeral_numeral_int,Uua)) ).
% ATP.lambda_43
tff(fact_6114_ATP_Olambda__44,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).
% ATP.lambda_44
tff(fact_6115_ATP_Olambda__45,axiom,
! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_do,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_bool_int(zero_n2684676970156552555ol_int,fNot(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).
% ATP.lambda_45
tff(fact_6116_ATP_Olambda__46,axiom,
! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_gv,Uu),Uua) = cons_nat(Uu,nat_list_decode(Uua)) ).
% ATP.lambda_46
tff(fact_6117_ATP_Olambda__47,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fy,Uu),Uua))
<=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).
% ATP.lambda_47
tff(fact_6118_ATP_Olambda__48,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bg,Uu),Uua))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).
% ATP.lambda_48
tff(fact_6119_ATP_Olambda__49,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fs,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).
% ATP.lambda_49
tff(fact_6120_ATP_Olambda__50,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bh,Uu),Uua))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).
% ATP.lambda_50
tff(fact_6121_ATP_Olambda__51,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).
% ATP.lambda_51
tff(fact_6122_ATP_Olambda__52,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_dg(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uua),Uu) ).
% ATP.lambda_52
tff(fact_6123_ATP_Olambda__53,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).
% ATP.lambda_53
tff(fact_6124_ATP_Olambda__54,axiom,
! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_df(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).
% ATP.lambda_54
tff(fact_6125_ATP_Olambda__55,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cs,Uu),Uua))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).
% ATP.lambda_55
tff(fact_6126_ATP_Olambda__56,axiom,
! [Uu: int,Uua: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cx,Uu),Uua))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).
% ATP.lambda_56
tff(fact_6127_ATP_Olambda__57,axiom,
! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ed,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).
% ATP.lambda_57
tff(fact_6128_ATP_Olambda__58,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).
% ATP.lambda_58
tff(fact_6129_ATP_Olambda__59,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_ag(Uu),Uua))
<=> member8440522571783428010at_nat(Uua,Uu) ) ).
% ATP.lambda_59
tff(fact_6130_ATP_Olambda__60,axiom,
! [Uu: set_list_nat,Uua: list_nat] :
( pp(aa_list_nat_bool(aTP_Lamm_ah(Uu),Uua))
<=> member_list_nat(Uua,Uu) ) ).
% ATP.lambda_60
tff(fact_6131_ATP_Olambda__61,axiom,
! [Uu: set_se6260736226359567993nt_int,Uua: set_Pr958786334691620121nt_int] :
( pp(aa_set7317184552576006522t_bool(aTP_Lamm_ai(Uu),Uua))
<=> member2340774599025711042nt_int(Uua,Uu) ) ).
% ATP.lambda_61
tff(fact_6132_ATP_Olambda__62,axiom,
! [Uu: set_set_nat,Uua: set_nat] :
( pp(aa_set_nat_bool(aTP_Lamm_aj(Uu),Uua))
<=> member_set_nat(Uua,Uu) ) ).
% ATP.lambda_62
tff(fact_6133_ATP_Olambda__63,axiom,
! [Uu: set_nat,Uua: nat] :
( pp(aa_nat_bool(aTP_Lamm_ak(Uu),Uua))
<=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).
% ATP.lambda_63
tff(fact_6134_ATP_Olambda__64,axiom,
! [Uu: set_int,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_al(Uu),Uua))
<=> member_int(Uua,Uu) ) ).
% ATP.lambda_64
tff(fact_6135_ATP_Olambda__65,axiom,
! [Uu: fun_nat_bool,Uua: nat] :
( pp(aa_nat_bool(aTP_Lamm_ge(Uu),Uua))
<=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).
% ATP.lambda_65
tff(fact_6136_ATP_Olambda__66,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_dt(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).
% ATP.lambda_66
tff(fact_6137_ATP_Olambda__67,axiom,
! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_gd,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_gc(Uu),Uua)) ).
% ATP.lambda_67
tff(fact_6138_ATP_Olambda__68,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_el,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ek(Uu),Uua)) ).
% ATP.lambda_68
tff(fact_6139_ATP_Olambda__69,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ej,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ei(Uu),Uua)) ).
% ATP.lambda_69
tff(fact_6140_ATP_Olambda__70,axiom,
! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_eh,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_eg(Uu),Uua)) ).
% ATP.lambda_70
tff(fact_6141_ATP_Olambda__71,axiom,
! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_ef,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ee(Uu),Uua)) ).
% ATP.lambda_71
tff(fact_6142_ATP_Olambda__72,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ec,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_eb(Uu),Uua)) ).
% ATP.lambda_72
tff(fact_6143_ATP_Olambda__73,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ce,Uu),Uua))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).
% ATP.lambda_73
tff(fact_6144_ATP_Olambda__74,axiom,
! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_gs(Uu),Uua) = produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uua),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Uu),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ).
% ATP.lambda_74
tff(fact_6145_ATP_Olambda__75,axiom,
! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_bv(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_75
tff(fact_6146_ATP_Olambda__76,axiom,
! [Uu: code_natural,Uua: nat] : aa_nat5282697797315388251atural(aTP_Lamm_bz(Uu),Uua) = aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uu),aa_nat_Code_natural(semiri3763490453095760265atural,Uua))) ).
% ATP.lambda_76
tff(fact_6147_ATP_Olambda__77,axiom,
! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_bx(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).
% ATP.lambda_77
tff(fact_6148_ATP_Olambda__78,axiom,
! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_by(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_78
tff(fact_6149_ATP_Olambda__79,axiom,
! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ca(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).
% ATP.lambda_79
tff(fact_6150_ATP_Olambda__80,axiom,
! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_bw(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).
% ATP.lambda_80
tff(fact_6151_ATP_Olambda__81,axiom,
! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_gu(Uu),Uua) = produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(modulo8411746178871703098atural,Uua),Uu)) ).
% ATP.lambda_81
tff(fact_6152_ATP_Olambda__82,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fi,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).
% ATP.lambda_82
tff(fact_6153_ATP_Olambda__83,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fj,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).
% ATP.lambda_83
tff(fact_6154_ATP_Olambda__84,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cy,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).
% ATP.lambda_84
tff(fact_6155_ATP_Olambda__85,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cz,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).
% ATP.lambda_85
tff(fact_6156_ATP_Olambda__86,axiom,
! [Uu: nat,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_gp(Uu),Uua) = some_P1914260805536162275it_nat(produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(Uua,Uu))) ).
% ATP.lambda_86
tff(fact_6157_ATP_Olambda__87,axiom,
! [Uu: fun_Pr2191293291710260995t_unit,Uua: product_unit] : aa_Pro1108561329362217960t_unit(aTP_Lamm_bc(Uu),Uua) = aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,aa_Pro8294289767544744956t_unit(Uu,Uua)) ).
% ATP.lambda_87
tff(fact_6158_ATP_Olambda__88,axiom,
! [Uu: fun_Product_unit_a,Uua: product_unit] : aa_Pro7886978635576167513Heap_a(aTP_Lamm_bb(Uu),Uua) = aa_a_H8829438773920479562Heap_a(heap_Time_return_a,aa_Product_unit_a(Uu,Uua)) ).
% ATP.lambda_88
tff(fact_6159_ATP_Olambda__89,axiom,
! [Uu: fun_a_a,Uua: a] : aa_a_H8829438773920479562Heap_a(aTP_Lamm_bd(Uu),Uua) = aa_a_H8829438773920479562Heap_a(heap_Time_return_a,aa_a_a(Uu,Uua)) ).
% ATP.lambda_89
tff(fact_6160_ATP_Olambda__90,axiom,
! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dp(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).
% ATP.lambda_90
tff(fact_6161_ATP_Olambda__91,axiom,
! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dq(Uu),Uua),Uub) = if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).
% ATP.lambda_91
tff(fact_6162_ATP_Olambda__92,axiom,
! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dr(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(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_92
tff(fact_6163_ATP_Olambda__93,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dz(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).
% ATP.lambda_93
tff(fact_6164_ATP_Olambda__94,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dx(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).
% ATP.lambda_94
tff(fact_6165_ATP_Olambda__95,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dy(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),Uub))) ).
% ATP.lambda_95
tff(fact_6166_ATP_Olambda__96,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_ea(Uu),Uua),Uub) = produc5135875973608410785r_bool(aa_Cod1783974652533396728nteger(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).
% ATP.lambda_96
tff(fact_6167_ATP_Olambda__97,axiom,
! [Uu: produc7822875418678951345atural,Uua: code_natural,Uub: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aTP_Lamm_gr(Uu),Uua),Uub) = produc8282080750456430313atural(aa_Cod8811915439870959971atural(aTP_Lamm_gq(Uua),Uub),produc6591912806276919810atural(aa_Pro7165575656943818355atural(next,Uu))) ).
% ATP.lambda_97
tff(fact_6168_ATP_Olambda__98,axiom,
! [Uu: bool,Uua: code_integer,Uub: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aTP_Lamm_gn(Uu),Uua),Uub) = produc7257819798349884575l_char(aa_boo1160955832611998748l_char(aTP_Lamm_gm(Uu),Uub),code_bit_cut_integer(Uua)) ).
% ATP.lambda_98
tff(fact_6169_ATP_Olambda__99,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ez(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ey(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_99
tff(fact_6170_ATP_Olambda__100,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ex(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ew(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_100
tff(fact_6171_ATP_Olambda__101,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ev(Uu),Uua),Uub))
<=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_eu(Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_101
tff(fact_6172_ATP_Olambda__102,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_et(Uu),Uua),Uub))
<=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_es(Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_102
tff(fact_6173_ATP_Olambda__103,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ep(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_eo(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_103
tff(fact_6174_ATP_Olambda__104,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_en(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_em(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_104
tff(fact_6175_ATP_Olambda__105,axiom,
! [Uu: nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fe(Uu),Uua),Uub))
<=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
& ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_105
tff(fact_6176_ATP_Olambda__106,axiom,
! [Uu: nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ff(Uu),Uua),Uub))
<=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
& ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).
% ATP.lambda_106
tff(fact_6177_ATP_Olambda__107,axiom,
! [Uu: nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fd(Uu),Uua),Uub))
<=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
& ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_107
tff(fact_6178_ATP_Olambda__108,axiom,
! [Uu: set_nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_da(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).
% ATP.lambda_108
tff(fact_6179_ATP_Olambda__109,axiom,
! [Uu: fun_Pr2191293291710260995t_unit,Uua: ref_Product_unit,Uub: product_unit] : aa_Pro1108561329362217960t_unit(aa_ref3209944945879839949t_unit(aTP_Lamm_bf(Uu),Uua),Uub) = heap_T2633723481742716231t_unit(aa_Pro1108561329362217960t_unit(ref_up627237114476368303t_unit(Uua),aa_Pro8294289767544744956t_unit(Uu,Uub)),aa_Pro5972434293396891430t_unit(aTP_Lamm_be(Uu),Uub)) ).
% ATP.lambda_109
tff(fact_6180_ATP_Olambda__110,axiom,
! [Uu: nat,Uua: nat,Uub: set_nat] :
( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_fx(Uu),Uua),Uub))
<=> ( member_set_nat(Uub,pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu)))
& ( finite_card_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_110
tff(fact_6181_ATP_Olambda__111,axiom,
! [Uu: set_nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_db(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).
% ATP.lambda_111
tff(fact_6182_ATP_Olambda__112,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ci(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).
% ATP.lambda_112
tff(fact_6183_ATP_Olambda__113,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dm(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).
% ATP.lambda_113
tff(fact_6184_ATP_Olambda__114,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ct(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).
% ATP.lambda_114
tff(fact_6185_ATP_Olambda__115,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dn(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).
% ATP.lambda_115
tff(fact_6186_ATP_Olambda__116,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cw(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).
% ATP.lambda_116
tff(fact_6187_ATP_Olambda__117,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cv(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).
% ATP.lambda_117
tff(fact_6188_ATP_Olambda__118,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cu(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).
% ATP.lambda_118
tff(fact_6189_ATP_Olambda__119,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fv(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).
% ATP.lambda_119
tff(fact_6190_ATP_Olambda__120,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fw(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uua))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uu)) ) ) ).
% ATP.lambda_120
tff(fact_6191_ATP_Olambda__121,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_at(Uu),Uua),Uub))
<=> ( member8440522571783428010at_nat(Uub,Uu)
& pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).
% ATP.lambda_121
tff(fact_6192_ATP_Olambda__122,axiom,
! [Uu: set_se6260736226359567993nt_int,Uua: fun_se3032438776331272065t_bool,Uub: set_Pr958786334691620121nt_int] :
( pp(aa_set7317184552576006522t_bool(aa_fun4116018846136612672t_bool(aTP_Lamm_au(Uu),Uua),Uub))
<=> ( member2340774599025711042nt_int(Uub,Uu)
& pp(aa_set7317184552576006522t_bool(Uua,Uub)) ) ) ).
% ATP.lambda_122
tff(fact_6193_ATP_Olambda__123,axiom,
! [Uu: set_set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_av(Uu),Uua),Uub))
<=> ( member_set_nat(Uub,Uu)
& pp(aa_set_nat_bool(Uua,Uub)) ) ) ).
% ATP.lambda_123
tff(fact_6194_ATP_Olambda__124,axiom,
! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_aw(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& pp(aa_nat_bool(Uua,Uub)) ) ) ).
% ATP.lambda_124
tff(fact_6195_ATP_Olambda__125,axiom,
! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ax(Uu),Uua),Uub))
<=> ( member_int(Uub,Uu)
& pp(aa_int_bool(Uua,Uub)) ) ) ).
% ATP.lambda_125
tff(fact_6196_ATP_Olambda__126,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fu(Uu),Uua),Uub))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).
% ATP.lambda_126
tff(fact_6197_ATP_Olambda__127,axiom,
! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_co(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(Uu,Uub))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).
% ATP.lambda_127
tff(fact_6198_ATP_Olambda__128,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cp(Uu),Uua),Uub))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).
% ATP.lambda_128
tff(fact_6199_ATP_Olambda__129,axiom,
! [Uu: fun_Pr292408699144966967t_unit,Uua: fun_Pr292408699144966967t_unit,Uub: product_unit] : aa_Pro1108561329362217960t_unit(aa_fun7583176353580103804t_unit(aTP_Lamm_ac(Uu),Uua),Uub) = heap_T2633723481742716231t_unit(aa_Pro1108561329362217960t_unit(Uu,Uub),Uua) ).
% ATP.lambda_129
tff(fact_6200_ATP_Olambda__130,axiom,
! [Uu: fun_Pr292408699144966967t_unit,Uua: fun_Pr5215307506330550056Heap_a,Uub: product_unit] : aa_Pro7886978635576167513Heap_a(aa_fun8407955803497444028Heap_a(aTP_Lamm_af(Uu),Uua),Uub) = heap_T757603679106148408unit_a(aa_Pro1108561329362217960t_unit(Uu,Uub),Uua) ).
% ATP.lambda_130
tff(fact_6201_ATP_Olambda__131,axiom,
! [Uu: fun_Pr5215307506330550056Heap_a,Uua: fun_a_2990225006182978344t_unit,Uub: product_unit] : aa_Pro1108561329362217960t_unit(aa_fun558819986861788875t_unit(aTP_Lamm_aa(Uu),Uua),Uub) = heap_T8685611227969916822t_unit(aa_Pro7886978635576167513Heap_a(Uu,Uub),Uua) ).
% ATP.lambda_131
tff(fact_6202_ATP_Olambda__132,axiom,
! [Uu: fun_Pr5215307506330550056Heap_a,Uua: fun_a_6093291656547882009Heap_a,Uub: product_unit] : aa_Pro7886978635576167513Heap_a(aa_fun7182454221404266251Heap_a(aTP_Lamm_ae(Uu),Uua),Uub) = heap_Time_bind_a_a(aa_Pro7886978635576167513Heap_a(Uu,Uub),Uua) ).
% ATP.lambda_132
tff(fact_6203_ATP_Olambda__133,axiom,
! [Uu: fun_a_2990225006182978344t_unit,Uua: fun_Pr5215307506330550056Heap_a,Uub: a] : aa_a_H8829438773920479562Heap_a(aa_fun2786189307152378797Heap_a(aTP_Lamm_ab(Uu),Uua),Uub) = heap_T757603679106148408unit_a(aa_a_H4862568901524997081t_unit(Uu,Uub),Uua) ).
% ATP.lambda_133
tff(fact_6204_ATP_Olambda__134,axiom,
! [Uu: fun_a_6093291656547882009Heap_a,Uua: fun_a_6093291656547882009Heap_a,Uub: a] : aa_a_H8829438773920479562Heap_a(aa_fun5994237636032447228Heap_a(aTP_Lamm_ad(Uu),Uua),Uub) = heap_Time_bind_a_a(aa_a_H8829438773920479562Heap_a(Uu,Uub),Uua) ).
% ATP.lambda_134
tff(fact_6205_ATP_Olambda__135,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).
% ATP.lambda_135
tff(fact_6206_ATP_Olambda__136,axiom,
! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gg(Uu),Uua),Uub))
<=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).
% ATP.lambda_136
tff(fact_6207_ATP_Olambda__137,axiom,
! [Uu: product_unit,Uua: fun_Pr292408699144966967t_unit,Uub: product_unit] : aa_Pro1108561329362217960t_unit(aa_fun7583176353580103804t_unit(aTP_Lamm_az(Uu),Uua),Uub) = aa_Pro1108561329362217960t_unit(Uua,Uu) ).
% ATP.lambda_137
tff(fact_6208_ATP_Olambda__138,axiom,
! [Uu: product_unit,Uua: fun_Pr5215307506330550056Heap_a,Uub: product_unit] : aa_Pro7886978635576167513Heap_a(aa_fun8407955803497444028Heap_a(aTP_Lamm_ay(Uu),Uua),Uub) = aa_Pro7886978635576167513Heap_a(Uua,Uu) ).
% ATP.lambda_138
tff(fact_6209_ATP_Olambda__139,axiom,
! [Uu: a,Uua: fun_a_6093291656547882009Heap_a,Uub: product_unit] : aa_Pro7886978635576167513Heap_a(aa_fun7182454221404266251Heap_a(aTP_Lamm_ba(Uu),Uua),Uub) = aa_a_H8829438773920479562Heap_a(Uua,Uu) ).
% ATP.lambda_139
tff(fact_6210_ATP_Olambda__140,axiom,
! [Uu: fun_Pr2191293291710260995t_unit,Uua: product_unit,Uub: product_unit] : aa_Pro1108561329362217960t_unit(aa_Pro5972434293396891430t_unit(aTP_Lamm_be(Uu),Uua),Uub) = aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,aa_Pro8294289767544744956t_unit(Uu,Uua)) ).
% ATP.lambda_140
tff(fact_6211_ATP_Olambda__141,axiom,
! [Uu: bool,Uua: bool,Uub: code_integer,Uuc: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aa_boo1160955832611998748l_char(aTP_Lamm_gm(Uu),Uua),Uub),Uuc) = produc7257819798349884575l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aTP_Lamm_gl(Uu),Uua),Uuc),code_bit_cut_integer(Uub)) ).
% ATP.lambda_141
tff(fact_6212_ATP_Olambda__142,axiom,
! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aa_Cod8811915439870959971atural(aTP_Lamm_gq(Uu),Uua),Uub),Uuc) = produc4480994950612372183atural(produc3574140220909816553atural(inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uu),Uuc),produc3574140220909816553atural(Uub,inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uua))) ).
% ATP.lambda_142
tff(fact_6213_ATP_Olambda__143,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_eb(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).
% ATP.lambda_143
tff(fact_6214_ATP_Olambda__144,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ck(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).
% ATP.lambda_144
tff(fact_6215_ATP_Olambda__145,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_cj(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).
% ATP.lambda_145
tff(fact_6216_ATP_Olambda__146,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_es(Uu),Uua),Uub),Uuc))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).
% ATP.lambda_146
tff(fact_6217_ATP_Olambda__147,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_eg(Uu),Uua),Uub),Uuc))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).
% ATP.lambda_147
tff(fact_6218_ATP_Olambda__148,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_eu(Uu),Uua),Uub),Uuc))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).
% ATP.lambda_148
tff(fact_6219_ATP_Olambda__149,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ee(Uu),Uua),Uub),Uuc))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).
% ATP.lambda_149
tff(fact_6220_ATP_Olambda__150,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ei(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).
% ATP.lambda_150
tff(fact_6221_ATP_Olambda__151,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ek(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_151
tff(fact_6222_ATP_Olambda__152,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_gc(Uu),Uua),Uub),Uuc))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).
% ATP.lambda_152
tff(fact_6223_ATP_Olambda__153,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_em(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_153
tff(fact_6224_ATP_Olambda__154,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_eo(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_154
tff(fact_6225_ATP_Olambda__155,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_155
tff(fact_6226_ATP_Olambda__156,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ew(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).
% ATP.lambda_156
tff(fact_6227_ATP_Olambda__157,axiom,
! [Uu: bool,Uua: bool,Uub: bool,Uuc: code_integer,Uud: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aTP_Lamm_gl(Uu),Uua),Uub),Uuc),Uud) = produc7257819798349884575l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aTP_Lamm_gk(Uu),Uua),Uub),Uud),code_bit_cut_integer(Uuc)) ).
% ATP.lambda_157
tff(fact_6228_ATP_Olambda__158,axiom,
! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: code_integer,Uue: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aTP_Lamm_gk(Uu),Uua),Uub),Uuc),Uud),Uue) = produc7257819798349884575l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aa_boo5194212545117631380l_char(aTP_Lamm_gj(Uu),Uua),Uub),Uuc),Uue),code_bit_cut_integer(Uud)) ).
% ATP.lambda_158
tff(fact_6229_ATP_Olambda__159,axiom,
! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: code_integer,Uuf: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aa_boo5194212545117631380l_char(aTP_Lamm_gj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc7257819798349884575l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aa_boo5194212545117631380l_char(aa_boo2023203282872912456l_char(aTP_Lamm_gi(Uu),Uua),Uub),Uuc),Uud),Uuf),code_bit_cut_integer(Uue)) ).
% ATP.lambda_159
tff(fact_6230_ATP_Olambda__160,axiom,
! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: code_integer] : aa_Cod6370812454140447392l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aa_boo5194212545117631380l_char(aa_boo2023203282872912456l_char(aa_boo9094958794169052458l_char(aTP_Lamm_gh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = char2(Uu,Uua,Uub,Uuc,Uud,Uue,Uuf) ).
% ATP.lambda_160
tff(fact_6231_ATP_Olambda__161,axiom,
! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: code_integer,Uug: bool] : aa_bool_char(aa_Cod6370812454140447392l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aa_boo5194212545117631380l_char(aa_boo2023203282872912456l_char(aTP_Lamm_gi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc7257819798349884575l_char(aa_boo1160955832611998748l_char(aa_boo3826374251381081086l_char(aa_boo3984127200019211826l_char(aa_boo5194212545117631380l_char(aa_boo2023203282872912456l_char(aa_boo9094958794169052458l_char(aTP_Lamm_gh(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),code_bit_cut_integer(Uuf)) ).
% ATP.lambda_161
tff(fact_6232_ATP_Olambda__162,axiom,
! [Uu: heap_T5738788834812785303t_unit,Uua: product_unit] : aa_Pro1108561329362217960t_unit(aTP_Lamm_an(Uu),Uua) = Uu ).
% ATP.lambda_162
tff(fact_6233_ATP_Olambda__163,axiom,
! [Uu: heap_Time_Heap_a,Uua: product_unit] : aa_Pro7886978635576167513Heap_a(aTP_Lamm_am(Uu),Uua) = Uu ).
% ATP.lambda_163
tff(fact_6234_ATP_Olambda__164,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cb,Uu),Uua))
<=> $true ) ).
% ATP.lambda_164
tff(fact_6235_ATP_Olambda__165,axiom,
! [Uu: int,Uua: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cc,Uu),Uua))
<=> $true ) ).
% ATP.lambda_165
tff(fact_6236_ATP_Olambda__166,axiom,
! [Uu: code_natural] : aa_Cod4272041906116450446atural(aTP_Lamm_bq,Uu) = Uu ).
% ATP.lambda_166
tff(fact_6237_ATP_Olambda__167,axiom,
! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_br,Uu) = Uu ).
% ATP.lambda_167
tff(fact_6238_ATP_Olambda__168,axiom,
! [Uu: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_gw,Uu) = Uu ).
% ATP.lambda_168
tff(fact_6239_ATP_Olambda__169,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_bs,Uu) = Uu ).
% ATP.lambda_169
tff(fact_6240_ATP_Olambda__170,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_bt,Uu) = Uu ).
% ATP.lambda_170
tff(fact_6241_ATP_Olambda__171,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_bu,Uu) = Uu ).
% ATP.lambda_171
tff(fact_6242_ATP_Olambda__172,axiom,
! [Uu: product_unit] : aa_Pro7886978635576167513Heap_a(aTP_Lamm_a,Uu) = m ).
% ATP.lambda_172
tff(fact_6243_ATP_Olambda__173,axiom,
! [Uu: code_natural] : aa_Cod4272041906116450446atural(aTP_Lamm_bl,Uu) = zero_z2226904508553997617atural ).
% ATP.lambda_173
tff(fact_6244_ATP_Olambda__174,axiom,
! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_bm,Uu) = zero_z3403309356797280102nteger ).
% ATP.lambda_174
tff(fact_6245_ATP_Olambda__175,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_bn,Uu) = zero_zero_rat ).
% ATP.lambda_175
tff(fact_6246_ATP_Olambda__176,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_bo,Uu) = zero_zero_nat ).
% ATP.lambda_176
tff(fact_6247_ATP_Olambda__177,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_bp,Uu) = zero_zero_int ).
% ATP.lambda_177
tff(fact_6248_ATP_Olambda__178,axiom,
! [Uu: nat] :
( pp(aa_nat_bool(aTP_Lamm_cq,Uu))
<=> $false ) ).
% ATP.lambda_178
tff(fact_6249_ATP_Olambda__179,axiom,
! [Uu: nat] :
( pp(aa_nat_bool(aTP_Lamm_cr,Uu))
<=> $true ) ).
% ATP.lambda_179
tff(fact_6250_ATP_Olambda__180,axiom,
! [Uu: bool] : aa_bool_fun_nat_nat(aTP_Lamm_bk,Uu) = suc ).
% ATP.lambda_180
tff(fact_6251_ATP_Olambda__181,axiom,
! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bi,Uu) = suc ).
% ATP.lambda_181
tff(fact_6252_ATP_Olambda__182,axiom,
! [Uu: int] : aa_int_fun_nat_nat(aTP_Lamm_bj,Uu) = suc ).
% ATP.lambda_182
% Helper facts (46)
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(fNot(P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(fNot(P))
| ~ pp(P) ) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fconj(P,Q))
| pp(Q) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fconj(P,Q))
| pp(P) ) ).
tff(help_fconj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(fconj(P,Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fdisj(P,Q))
| pp(P)
| pp(Q) ) ).
tff(help_fdisj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(Q)
| pp(fdisj(P,Q)) ) ).
tff(help_fdisj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| pp(fdisj(P,Q)) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] : if_int(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] : if_int(fTrue,X,Y) = X ).
tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).
tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).
tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] : aa_num_num(if_num(fFalse,X),Y) = Y ).
tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] : aa_num_num(if_num(fTrue,X),Y) = X ).
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( X != Y )
| pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( X != Y )
| pp(aa_num_bool(aa_num_fun_num_bool(fequal_num,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(fequal_num,X),Y))
| ( X = Y ) ) ).
tff(help_fChoice_1_1_fChoice_001t__Int__Oint_T,axiom,
! [P: fun_int_bool] : aa_int_bool(P,fChoice_int(P)) = aa_fun_int_bool_bool(fEx_int,P) ).
tff(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: fun_nat_bool] : aa_nat_bool(P,fChoice_nat(P)) = aa_fun_nat_bool_bool(fEx_nat,P) ).
tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
! [X: bool,Y: bool] :
( ( X != Y )
| pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
! [X: bool,Y: bool] :
( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
| ( X = Y ) ) ).
tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fFalse,X),Y) = Y ).
tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fTrue,X),Y) = X ).
tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
| ( X = Y ) ) ).
tff(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: fun_set_nat_bool] : aa_set_nat_bool(P,fChoice_set_nat(P)) = aa_fun5644405980846309865l_bool(fEx_set_nat,P) ).
tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_Itf__a_J_T,axiom,
! [X: heap_Time_Heap_a,Y: heap_Time_Heap_a] : if_Heap_Time_Heap_a(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_Itf__a_J_T,axiom,
! [X: heap_Time_Heap_a,Y: heap_Time_Heap_a] : if_Heap_Time_Heap_a(fTrue,X,Y) = X ).
tff(help_fequal_2_1_fequal_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_T,axiom,
! [X: fun_nat_bool,Y: fun_nat_bool] :
( ( X != Y )
| pp(aa_fun_nat_bool_bool(aa_fun1020848501848070430l_bool(fequal_fun_nat_bool,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_T,axiom,
! [X: fun_nat_bool,Y: fun_nat_bool] :
( ~ pp(aa_fun_nat_bool_bool(aa_fun1020848501848070430l_bool(fequal_fun_nat_bool,X),Y))
| ( X = Y ) ) ).
tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).
tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X,Y) = X ).
tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_T,axiom,
! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit] : if_Hea8138950348631371857t_unit(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_T,axiom,
! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit] : if_Hea8138950348631371857t_unit(fTrue,X,Y) = X ).
tff(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: fun_Pr7757163693017731481t_bool] : aa_Pro2232559814647311250t_bool(P,fChoic6978938873391328853at_nat(P)) = aa_fun8663595511526100224l_bool(fEx_Pr44684694416842691at_nat,P) ).
tff(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [P: fun_se3032438776331272065t_bool] : aa_set7317184552576006522t_bool(P,fChoic2749331249758569069nt_int(P)) = aa_fun2006528745882660120l_bool(fEx_se7899788091526085595nt_int,P) ).
tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).
% Conjectures (1)
tff(conj_0,conjecture,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_a(heap_T757603679106148408unit_a(refine_Imp_assert(p),aTP_Lamm_a),h)),t)) ).
%------------------------------------------------------------------------------